{
  "index": "1990-A-2",
  "type": "ANA",
  "tag": [
    "ANA",
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "$\\sqrt[3]{n} - \\sqrt[3]{m}$ ($n,m = 0, 1, 2, \\dots$)?",
  "solution": "Solution 1. By the binomial expansion,\n\\[\n\\sqrt[3]{n+1}-\\sqrt[3]{n}=n^{1 / 3}\\left(1+\\frac{1}{n}\\right)^{1 / 3}-n^{1 / 3}=n^{1 / 3}\\left(1+O\\left(\\frac{1}{n}\\right)\\right)-n^{1 / 3}=O\\left(n^{-2 / 3}\\right)\n\\]\nso \\( \\sqrt[3]{n+1}-\\sqrt[3]{n} \\rightarrow 0 \\) as \\( n \\rightarrow \\infty \\). (Alternatively, one could use\n\\[\n\\sqrt[3]{n+1}-\\sqrt[3]{n}=\\frac{1}{\\sqrt[3]{(n+1)^{2}}+\\sqrt[3]{n(n+1)}+\\sqrt[3]{n^{2}}}=O\\left(n^{-2 / 3}\\right)\n\\]\nor\n\\[\n\\sqrt[3]{n+1}-\\sqrt[3]{n}=\\frac{1}{3} \\int_{n}^{n+1} x^{-2 / 3} d x=O\\left(n^{-2 / 3}\\right)\n\\]\nor the Mean Value Theorem applied to the difference quotient \\( \\frac{\\sqrt[3]{n+1}-\\sqrt[3]{n}}{(n+1)-n} \\).)\nIf \\( m>r^{3} \\), then the numbers\n\\[\n0=\\sqrt[3]{m}-\\sqrt[3]{m}<\\sqrt[3]{m+1}-\\sqrt[3]{m}<\\cdots<\\sqrt[3]{m+7 m}-\\sqrt[3]{m}=\\sqrt[3]{m}\n\\]\npartition the interval \\( [0, \\sqrt[3]{m}] \\), containing \\( r \\), in such a way that the largest subinterval is of size \\( O\\left(m^{-2 / 3}\\right) \\). By taking \\( m \\) sufficiently large, one can find a difference among these that is arbitrarily close to \\( r \\).\n\nRemark. We show more generally, that for any sequence \\( \\left\\{a_{n}\\right\\} \\) with \\( a_{n} \\rightarrow+\\infty \\) and \\( a_{n+1}-a_{n} \\rightarrow 0 \\), the set \\( S=\\left\\{a_{n}-a_{m}: m, n \\geq 0\\right\\} \\) is dense in \\( \\mathbb{R} \\). Given \\( r \\geq 0 \\) and \\( \\epsilon>0 \\), fix \\( m \\) such that \\( \\left|a_{M+1}-a_{M}\\right|<\\epsilon \\) for all \\( M \\geq m \\). If \\( n \\) is the smallest integer \\( \\geq m \\) with \\( a_{n} \\geq a_{m}+r \\), then \\( a_{n}<a_{m}+r+\\epsilon \\), so \\( a_{n}-a_{m} \\) is within \\( \\epsilon \\) of \\( r \\). This shows that \\( S \\) is dense in \\( [0, \\infty) \\), and by symmetry \\( S \\) is dense also in \\( (-\\infty, 0] \\).\n\nRemark. Let \\( f(x) \\) be a function such that \\( f(x) \\rightarrow+\\infty \\) and \\( f^{\\prime}(x) \\rightarrow 0 \\) as \\( x \\rightarrow+\\infty \\). The Mean Value Theorem shows that the hypotheses of the previous remark are satisfied by the sequence \\( a_{n}=f(n) \\).\n\nSolution 2. Fix \\( r \\in \\mathbb{R} \\) and \\( \\epsilon>0 \\). Then for sufficiently large positive integers \\( n \\),\n\\[\n(n+r)^{3}-(n+r-\\epsilon)^{3}=3 n^{2} \\epsilon+O(n)>1\n\\]\nso \\( (n+r-\\epsilon)^{3} \\leq\\left\\lfloor(n+r)^{3}\\right\\rfloor \\leq(n+r)^{3} \\), and \\( \\sqrt[3]{\\left\\lfloor(n+r)^{3}\\right\\rfloor} \\) is within \\( \\epsilon \\) of \\( n+r \\). Hence\n\\[\n\\lim _{n \\rightarrow \\infty}\\left(\\sqrt[3]{\\left\\lfloor(n+r)^{3}\\right\\rfloor}-\\sqrt[3]{n}\\right)=r\n\\]\n\nSolution 3. As in Solution \\( 1, \\sqrt[3]{n+1}-\\sqrt[3]{n} \\rightarrow 0 \\) as \\( n \\rightarrow \\infty \\), so the set \\( S=\\{\\sqrt[3]{n}-\\sqrt[3]{m}\\} \\) contains arbitrarily small positive numbers. Also \\( S \\) is closed under multiplication by positive integers \\( k \\) since \\( k(\\sqrt[3]{n}-\\sqrt[3]{m})=\\sqrt[3]{k^{3} n}-\\sqrt[3]{k^{3} m} \\). Any set with the preceding two properties is dense in \\( [0, \\infty) \\), because any finite open interval \\( (a, b) \\) in \\( [0, \\infty) \\) contains a multiple of any element of \\( S \\cap(0, b-a) \\). By symmetry \\( S \\) is dense in \\( (-\\infty, 0] \\) too.\n\nSolution 4. Let \\( b_{n}=(n \\sqrt[3]{5} \\bmod 1) \\in[0,1] \\). As in the remark following 1988B3, \\( \\left\\{b_{n}\\right\\} \\) is dense in \\( [0,1] \\). Thus given \\( \\epsilon>0 \\), we can find \\( n \\) such that \\( n \\sqrt[3]{5}-r \\) is within \\( \\epsilon \\) of some integer \\( m \\geq 0 \\). Then \\( \\sqrt[3]{5 n^{3}}-\\sqrt[3]{m^{3}}=n \\sqrt[3]{5}-m \\) is within \\( \\epsilon \\) of \\( r \\).",
  "vars": [
    "n",
    "m",
    "x",
    "M",
    "k",
    "a_n",
    "a_m",
    "a_M",
    "b_n",
    "f",
    "S"
  ],
  "params": [
    "r",
    "\\\\epsilon"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "indexer",
        "m": "counter",
        "x": "coordinate",
        "M": "bigindex",
        "k": "multfact",
        "a_n": "seqtermn",
        "a_m": "seqtermm",
        "a_M": "seqtermbig",
        "b_n": "seqbterm",
        "f": "funcmain",
        "S": "denseset",
        "r": "desired",
        "\\epsilon": "tolerance"
      },
      "question": "$\\sqrt[3]{indexer} - \\sqrt[3]{counter}$ ($indexer, counter = 0, 1, 2, \\dots$)?",
      "solution": "Solution 1. By the binomial expansion,\n\\[\n\\sqrt[3]{indexer+1}-\\sqrt[3]{indexer}=indexer^{1 / 3}\\left(1+\\frac{1}{indexer}\\right)^{1 / 3}-indexer^{1 / 3}=indexer^{1 / 3}\\left(1+O\\left(\\frac{1}{indexer}\\right)\\right)-indexer^{1 / 3}=O\\left(indexer^{-2 / 3}\\right)\n\\]\nso \\( \\sqrt[3]{indexer+1}-\\sqrt[3]{indexer} \\rightarrow 0 \\) as \\( indexer \\rightarrow \\infty \\). (Alternatively, one could use\n\\[\n\\sqrt[3]{indexer+1}-\\sqrt[3]{indexer}=\\frac{1}{\\sqrt[3]{(indexer+1)^{2}}+\\sqrt[3]{indexer(indexer+1)}+\\sqrt[3]{indexer^{2}}}=O\\left(indexer^{-2 / 3}\\right)\n\\]\nor\n\\[\n\\sqrt[3]{indexer+1}-\\sqrt[3]{indexer}=\\frac{1}{3} \\int_{indexer}^{indexer+1} coordinate^{-2 / 3}\\,d coordinate=O\\left(indexer^{-2 / 3}\\right)\n\\]\nor the Mean Value Theorem applied to the difference quotient \\( \\frac{\\sqrt[3]{indexer+1}-\\sqrt[3]{indexer}}{(indexer+1)-indexer} \\).)\nIf \\( counter>desired^{3} \\), then the numbers\n\\[\n0=\\sqrt[3]{counter}-\\sqrt[3]{counter}<\\sqrt[3]{counter+1}-\\sqrt[3]{counter}<\\cdots<\\sqrt[3]{counter+7\\,counter}-\\sqrt[3]{counter}=\\sqrt[3]{counter}\n\\]\npartition the interval \\( [0, \\sqrt[3]{counter}] \\), containing \\( desired \\), in such a way that the largest subinterval is of size \\( O\\left(counter^{-2 / 3}\\right) \\). By taking \\( counter \\) sufficiently large, one can find a difference among these that is arbitrarily close to \\( desired \\).\n\nRemark. We show more generally that for any sequence \\( \\{seqtermn\\} \\) with \\( seqtermn \\rightarrow +\\infty \\) and \\( a_{indexer+1}-seqtermn \\rightarrow 0 \\), the set \\( denseset=\\{seqtermn-seqtermm:\\,counter, indexer \\ge 0\\} \\) is dense in \\( \\mathbb{R} \\). Given \\( desired \\ge 0 \\) and \\( tolerance>0 \\), fix \\( counter \\) such that \\( |a_{bigindex+1}-seqtermbig|<tolerance \\) for all \\( bigindex \\ge counter \\). If \\( indexer \\) is the smallest integer \\( \\ge counter \\) with \\( a_{indexer} \\ge seqtermm+desired \\), then \\( a_{indexer}<seqtermm+desired+tolerance \\), so \\( a_{indexer}-seqtermm \\) is within \\( tolerance \\) of \\( desired \\). This shows that \\( denseset \\) is dense in \\( [0,\\infty) \\); by symmetry it is dense also in \\( (-\\infty,0] \\).\n\nRemark. Let \\( funcmain(coordinate) \\) be a function such that \\( funcmain(coordinate) \\rightarrow +\\infty \\) and \\( funcmain^{\\prime}(coordinate) \\rightarrow 0 \\) as \\( coordinate \\rightarrow +\\infty \\). The Mean Value Theorem shows that the hypotheses of the previous remark are satisfied by the sequence \\( seqtermn=funcmain(indexer) \\).\n\nSolution 2. Fix \\( desired \\in \\mathbb{R} \\) and \\( tolerance>0 \\). For sufficiently large positive integers \\( indexer \\),\n\\[\n(indexer+desired)^{3}-(indexer+desired-tolerance)^{3}=3\\,indexer^{2}\\,tolerance+O(indexer)>1,\n\\]\nso \\( (indexer+desired-tolerance)^{3}\\le \\lfloor(indexer+desired)^{3}\\rfloor \\le (indexer+desired)^{3} \\), and \\( \\sqrt[3]{\\lfloor(indexer+desired)^{3}\\rfloor} \\) is within \\( tolerance \\) of \\( indexer+desired \\). Hence\n\\[\n\\lim_{indexer\\to\\infty}\\bigl(\\sqrt[3]{\\lfloor(indexer+desired)^{3}\\rfloor}-\\sqrt[3]{indexer}\\bigr)=desired.\n\\]\n\nSolution 3. As in Solution 1, \\( \\sqrt[3]{indexer+1}-\\sqrt[3]{indexer}\\to0 \\) as \\( indexer\\to\\infty \\), so the set \\( denseset=\\{\\sqrt[3]{indexer}-\\sqrt[3]{counter}\\} \\) contains arbitrarily small positive numbers. Also \\( denseset \\) is closed under multiplication by positive integers \\( multfact \\), since\n\\[\nmultfact\\bigl(\\sqrt[3]{indexer}-\\sqrt[3]{counter}\\bigr)=\\sqrt[3]{multfact^{3}indexer}-\\sqrt[3]{multfact^{3}counter}.\n\\]\nAny set with these two properties is dense in \\( [0,\\infty) \\), because any finite open interval \\( (a,b) \\subset [0,\\infty) \\) contains a multiple of an element of \\( denseset\\cap(0,b-a) \\). By symmetry, \\( denseset \\) is dense in \\( (-\\infty,0] \\) as well.\n\nSolution 4. Let \\( seqbterm=(indexer\\sqrt[3]{5}\\bmod1)\\in[0,1] \\). As in the remark following 1988B3, \\( \\{seqbterm\\} \\) is dense in \\( [0,1] \\). Thus, given \\( tolerance>0 \\), we can find \\( indexer \\) such that \\( indexer\\sqrt[3]{5}-desired \\) is within \\( tolerance \\) of some integer \\( counter\\ge0 \\). Then\n\\[\n\\sqrt[3]{5\\,indexer^{3}}-\\sqrt[3]{counter^{3}}=indexer\\sqrt[3]{5}-counter\n\\]\nis within \\( tolerance \\) of \\( desired \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "marigolds",
        "m": "skylarkic",
        "x": "tangerine",
        "M": "windflower",
        "k": "porcupine",
        "a_n": "buttercups",
        "a_m": "caterpillar",
        "a_M": "dandelions",
        "b_n": "hummingbird",
        "f": "watercress",
        "S": "pebblerock",
        "r": "rainshadow",
        "\\epsilon": "lavendereps"
      },
      "question": "$\\sqrt[3]{marigolds} - \\sqrt[3]{skylarkic}$ ($marigolds,skylarkic = 0, 1, 2, \\dots$)?",
      "solution": "Solution 1. By the binomial expansion,\n\\[\n\\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds}=marigolds^{1 / 3}\\left(1+\\frac{1}{marigolds}\\right)^{1 / 3}-marigolds^{1 / 3}=marigolds^{1 / 3}\\left(1+O\\left(\\frac{1}{marigolds}\\right)\\right)-marigolds^{1 / 3}=O\\left(marigolds^{-2 / 3}\\right)\n\\]\nso \\( \\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds} \\rightarrow 0 \\) as \\( marigolds \\rightarrow \\infty \\). (Alternatively, one could use\n\\[\n\\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds}=\\frac{1}{\\sqrt[3]{(marigolds+1)^{2}}+\\sqrt[3]{marigolds(marigolds+1)}+\\sqrt[3]{marigolds^{2}}}=O\\left(marigolds^{-2 / 3}\\right)\n\\]\nor\n\\[\n\\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds}=\\frac{1}{3} \\int_{marigolds}^{marigolds+1} tangerine^{-2 / 3} d tangerine=O\\left(marigolds^{-2 / 3}\\right)\n\\]\nor the Mean Value Theorem applied to the difference quotient \\( \\frac{\\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds}}{(marigolds+1)-marigolds} \\).)\nIf \\( skylarkic>rainshadow^{3} \\), then the numbers\n\\[\n0=\\sqrt[3]{skylarkic}-\\sqrt[3]{skylarkic}<\\sqrt[3]{skylarkic+1}-\\sqrt[3]{skylarkic}<\\cdots<\\sqrt[3]{skylarkic+7 skylarkic}-\\sqrt[3]{skylarkic}=\\sqrt[3]{skylarkic}\n\\]\npartition the interval \\( [0, \\sqrt[3]{skylarkic}] \\), containing \\( rainshadow \\), in such a way that the largest subinterval is of size \\( O\\left(skylarkic^{-2 / 3}\\right) \\). By taking \\( skylarkic \\) sufficiently large, one can find a difference among these that is arbitrarily close to \\( rainshadow \\).\n\nRemark. We show more generally, that for any sequence \\( \\{buttercups\\} \\) with \\( buttercups \\rightarrow+\\infty \\) and \\( a_{marigolds+1}-buttercups \\rightarrow 0 \\), the set \\( pebblerock=\\{buttercups-caterpillar: skylarkic, marigolds \\geq 0\\} \\) is dense in \\( \\mathbb{R} \\). Given \\( rainshadow \\geq 0 \\) and \\( lavendereps>0 \\), fix \\( skylarkic \\) such that \\( |a_{windflower+1}-dandelions|<lavendereps \\) for all \\( windflower \\geq skylarkic \\). If \\( marigolds \\) is the smallest integer \\( \\geq skylarkic \\) with \\( a_{marigolds} \\geq a_{skylarkic}+rainshadow \\), then \\( a_{marigolds}<a_{skylarkic}+rainshadow+lavendereps \\), so \\( a_{marigolds}-a_{skylarkic} \\) is within \\( lavendereps \\) of \\( rainshadow \\). This shows that \\( pebblerock \\) is dense in \\( [0, \\infty) \\), and by symmetry \\( pebblerock \\) is dense also in \\( (-\\infty, 0] \\).\n\nRemark. Let \\( watercress(tangerine) \\) be a function such that \\( watercress(tangerine) \\rightarrow+\\infty \\) and \\( watercress^{\\prime}(tangerine) \\rightarrow 0 \\) as \\( tangerine \\rightarrow+\\infty \\). The Mean Value Theorem shows that the hypotheses of the previous remark are satisfied by the sequence \\( buttercups=watercress(marigolds) \\).\n\nSolution 2. Fix \\( rainshadow \\in \\mathbb{R} \\) and \\( lavendereps>0 \\). Then for sufficiently large positive integers \\( marigolds \\),\n\\[\n(marigolds+rainshadow)^{3}-(marigolds+rainshadow-lavendereps)^{3}=3 marigolds^{2} lavendereps+O(marigolds)>1\n\\]\nso \\( (marigolds+rainshadow-lavendereps)^{3} \\leq\\lfloor(marigolds+rainshadow)^{3}\\rfloor \\leq(marigolds+rainshadow)^{3} \\), and \\( \\sqrt[3]{\\lfloor(marigolds+rainshadow)^{3}\\rfloor} \\) is within \\( lavendereps \\) of \\( marigolds+rainshadow \\). Hence\n\\[\n\\lim_{marigolds \\rightarrow \\infty}\\left(\\sqrt[3]{\\lfloor(marigolds+rainshadow)^{3}\\rfloor}-\\sqrt[3]{marigolds}\\right)=rainshadow\n\\]\n\nSolution 3. As in Solution 1, \\( \\sqrt[3]{marigolds+1}-\\sqrt[3]{marigolds} \\rightarrow 0 \\) as \\( marigolds \\rightarrow \\infty \\), so the set \\( pebblerock=\\{\\sqrt[3]{marigolds}-\\sqrt[3]{skylarkic}\\} \\) contains arbitrarily small positive numbers. Also \\( pebblerock \\) is closed under multiplication by positive integers \\( porcupine \\) since \\( porcupine(\\sqrt[3]{marigolds}-\\sqrt[3]{skylarkic})=\\sqrt[3]{porcupine^{3} marigolds}-\\sqrt[3]{porcupine^{3} skylarkic} \\). Any set with the preceding two properties is dense in \\( [0, \\infty) \\), because any finite open interval \\( (a, b) \\) in \\( [0, \\infty) \\) contains a multiple of any element of \\( pebblerock \\cap(0, b-a) \\). By symmetry \\( pebblerock \\) is dense in \\( (-\\infty, 0] \\) too.\n\nSolution 4. Let \\( hummingbird=(marigolds \\sqrt[3]{5} \\bmod 1) \\in[0,1] \\). As in the remark following 1988B3, \\( \\{hummingbird\\} \\) is dense in \\( [0,1] \\). Thus given \\( lavendereps>0 \\), we can find \\( marigolds \\) such that \\( marigolds \\sqrt[3]{5}-rainshadow \\) is within \\( lavendereps \\) of some integer \\( skylarkic \\geq 0 \\). Then \\( \\sqrt[3]{5 marigolds^{3}}-\\sqrt[3]{skylarkic^{3}}=marigolds \\sqrt[3]{5}-skylarkic \\) is within \\( lavendereps \\) of \\( rainshadow \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "fixedconstant",
        "m": "staticvalue",
        "x": "stationarypoint",
        "M": "settledindex",
        "k": "unwaveringcount",
        "a_n": "constantsequence",
        "a_m": "steadysequence",
        "a_M": "stablesequence",
        "b_n": "stagnantseries",
        "f": "stagnantmap",
        "S": "sparsecollection",
        "r": "fixedreference",
        "\\\\epsilon": "hugeerror"
      },
      "question": "$\\sqrt[3]{fixedconstant} - \\sqrt[3]{staticvalue}$ ($fixedconstant,staticvalue = 0, 1, 2, \\dots$)?",
      "solution": "Solution 1. By the binomial expansion,\n\\[\n\\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant}=fixedconstant^{1 / 3}\\left(1+\\frac{1}{fixedconstant}\\right)^{1 / 3}-fixedconstant^{1 / 3}=fixedconstant^{1 / 3}\\left(1+O\\left(\\frac{1}{fixedconstant}\\right)\\right)-fixedconstant^{1 / 3}=O\\left(fixedconstant^{-2 / 3}\\right)\n\\]\nso \\( \\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant} \\rightarrow 0 \\) as \\( fixedconstant \\rightarrow \\infty \\). (Alternatively, one could use\n\\[\n\\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant}=\\frac{1}{\\sqrt[3]{(fixedconstant+1)^{2}}+\\sqrt[3]{fixedconstant(fixedconstant+1)}+\\sqrt[3]{fixedconstant^{2}}}=O\\left(fixedconstant^{-2 / 3}\\right)\n\\]\nor\n\\[\n\\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant}=\\frac{1}{3} \\int_{fixedconstant}^{fixedconstant+1} stationarypoint^{-2 / 3} \\, d\\, stationarypoint=O\\left(fixedconstant^{-2 / 3}\\right)\n\\]\nor the Mean Value Theorem applied to the difference quotient \\( \\frac{\\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant}}{(fixedconstant+1)-fixedconstant} \\).)\nIf \\( staticvalue>fixedreference^{3} \\), then the numbers\n\\[\n0=\\sqrt[3]{staticvalue}-\\sqrt[3]{staticvalue}<\\sqrt[3]{staticvalue+1}-\\sqrt[3]{staticvalue}<\\cdots<\\sqrt[3]{staticvalue+7 staticvalue}-\\sqrt[3]{staticvalue}=\\sqrt[3]{staticvalue}\n\\]\npartition the interval \\( [0, \\sqrt[3]{staticvalue}] \\), containing \\( fixedreference \\), in such a way that the largest subinterval is of size \\( O\\left(staticvalue^{-2 / 3}\\right) \\). By taking \\( staticvalue \\) sufficiently large, one can find a difference among these that is arbitrarily close to \\( fixedreference \\).\n\nRemark. We show more generally that for any sequence \\( \\{constantsequence\\} \\) with \\( constantsequence \\rightarrow +\\infty \\) and \\( constantsequenceplusone-constantsequence \\rightarrow 0 \\), the set \\( sparsecollection=\\{constantsequence-steadysequence: staticvalue,fixedconstant \\ge 0\\} \\) is dense in \\( \\mathbb{R} \\). Given \\( fixedreference \\ge 0 \\) and \\( hugeerror>0 \\), fix \\( staticvalue \\) such that \\( |stablesequenceplusone-stablesequence|<hugeerror \\) for all \\( settledindex \\ge staticvalue \\). If \\( fixedconstant \\) is the smallest integer \\( \\ge staticvalue \\) with \\( constantsequence \\ge steadysequence+fixedreference \\), then \\( constantsequence<steadysequence+fixedreference+hugeerror \\), so \\( constantsequence-steadysequence \\) is within \\( hugeerror \\) of \\( fixedreference \\). This shows that \\( sparsecollection \\) is dense in \\( [0,\\infty) \\), and by symmetry \\( sparsecollection \\) is dense also in \\( (-\\infty,0] \\).\n\nRemark. Let \\( stagnantmap(stationarypoint) \\) be a function such that \\( stagnantmap(stationarypoint) \\rightarrow +\\infty \\) and \\( stagnantmap' (stationarypoint) \\rightarrow 0 \\) as \\( stationarypoint \\rightarrow +\\infty \\). The Mean Value Theorem shows that the hypotheses of the previous remark are satisfied by the sequence \\( constantsequence=stagnantmap(fixedconstant) \\).\n\nSolution 2. Fix \\( fixedreference \\in \\mathbb{R} \\) and \\( hugeerror>0 \\). Then for sufficiently large positive integers \\( fixedconstant \\),\n\\[\n(fixedconstant+fixedreference)^{3}-(fixedconstant+fixedreference-hugeerror)^{3}=3 fixedconstant^{2} hugeerror+O(fixedconstant)>1\n\\]\nso \\( (fixedconstant+fixedreference-hugeerror)^{3} \\le \\lfloor(fixedconstant+fixedreference)^{3}\\rfloor \\le (fixedconstant+fixedreference)^{3} \\), and \\( \\sqrt[3]{\\lfloor(fixedconstant+fixedreference)^{3}\\rfloor} \\) is within \\( hugeerror \\) of \\( fixedconstant+fixedreference \\). Hence\n\\[\n\\lim_{fixedconstant \\to \\infty}\\Bigl(\\sqrt[3]{\\lfloor(fixedconstant+fixedreference)^{3}\\rfloor}-\\sqrt[3]{fixedconstant}\\Bigr)=fixedreference.\n\\]\n\nSolution 3. As in Solution 1, \\( \\sqrt[3]{fixedconstant+1}-\\sqrt[3]{fixedconstant} \\rightarrow 0 \\) as \\( fixedconstant \\rightarrow \\infty \\), so the set \\( sparsecollection=\\{\\sqrt[3]{fixedconstant}-\\sqrt[3]{staticvalue}\\} \\) contains arbitrarily small positive numbers. Also \\( sparsecollection \\) is closed under multiplication by positive integers \\( unwaveringcount \\) since \\( unwaveringcount(\\sqrt[3]{fixedconstant}-\\sqrt[3]{staticvalue})=\\sqrt[3]{unwaveringcount^{3} fixedconstant}-\\sqrt[3]{unwaveringcount^{3} staticvalue} \\). Any set with the preceding two properties is dense in \\( [0,\\infty) \\), because any finite open interval \\( (a,b) \\) in \\( [0,\\infty) \\) contains a multiple of any element of \\( sparsecollection \\cap (0,b-a) \\). By symmetry \\( sparsecollection \\) is dense in \\( (-\\infty,0] \\) too.\n\nSolution 4. Let \\( stagnantseries=(fixedconstant \\sqrt[3]{5} \\bmod 1) \\in [0,1] \\). As in the remark following 1988B3, \\( \\{stagnantseries\\} \\) is dense in \\([0,1]\\). Thus given \\( hugeerror>0 \\), we can find \\( fixedconstant \\) such that \\( fixedconstant \\sqrt[3]{5}-fixedreference \\) is within \\( hugeerror \\) of some integer \\( staticvalue \\ge 0 \\). Then \\( \\sqrt[3]{5 fixedconstant^{3}}-\\sqrt[3]{staticvalue^{3}}=fixedconstant \\sqrt[3]{5}-staticvalue \\) is within \\( hugeerror \\) of \\( fixedreference \\)."
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "m": "hjgrksla",
        "x": "vldqmpre",
        "M": "tzbnwrce",
        "k": "psqdlmva",
        "a_n": "fjslqwer",
        "a_m": "gnhsdapi",
        "a_M": "kzptynou",
        "b_n": "wmpqzeru",
        "f": "ojxlerpu",
        "S": "nhaqtwes",
        "r": "lbzgvhio",
        "\\epsilon": "aytksrnd"
      },
      "question": "$\\sqrt[3]{qzxwvtnp} - \\sqrt[3]{hjgrksla}$ ($qzxwvtnp,hjgrksla = 0, 1, 2, \\dots$)?",
      "solution": "Solution 1. By the binomial expansion,\n\\[\n\\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp}=qzxwvtnp^{1 / 3}\\left(1+\\frac{1}{qzxwvtnp}\\right)^{1 / 3}-qzxwvtnp^{1 / 3}=qzxwvtnp^{1 / 3}\\left(1+O\\left(\\frac{1}{qzxwvtnp}\\right)\\right)-qzxwvtnp^{1 / 3}=O\\left(qzxwvtnp^{-2 / 3}\\right)\n\\]\nso \\( \\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp} \\rightarrow 0 \\) as \\( qzxwvtnp \\rightarrow \\infty \\). (Alternatively, one could use\n\\[\n\\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp}=\\frac{1}{\\sqrt[3]{(qzxwvtnp+1)^{2}}+\\sqrt[3]{qzxwvtnp(qzxwvtnp+1)}+\\sqrt[3]{qzxwvtnp^{2}}}=O\\left(qzxwvtnp^{-2 / 3}\\right)\n\\]\nor\n\\[\n\\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp}=\\frac{1}{3} \\int_{qzxwvtnp}^{qzxwvtnp+1} vldqmpre^{-2 / 3} d vldqmpre=O\\left(qzxwvtnp^{-2 / 3}\\right)\n\\]\nor the Mean Value Theorem applied to the difference quotient \\( \\frac{\\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp}}{(qzxwvtnp+1)-qzxwvtnp} \\).)\nIf \\( hjgrksla>lbzgvhio^{3} \\), then the numbers\n\\[\n0=\\sqrt[3]{hjgrksla}-\\sqrt[3]{hjgrksla}<\\sqrt[3]{hjgrksla+1}-\\sqrt[3]{hjgrksla}<\\cdots<\\sqrt[3]{hjgrksla+7 hjgrksla}-\\sqrt[3]{hjgrksla}=\\sqrt[3]{hjgrksla}\n\\]\npartition the interval \\( [0, \\sqrt[3]{hjgrksla}] \\), containing \\( lbzgvhio \\), in such a way that the largest subinterval is of size \\( O\\left(hjgrksla^{-2 / 3}\\right) \\). By taking \\( hjgrksla \\) sufficiently large, one can find a difference among these that is arbitrarily close to \\( lbzgvhio \\).\n\nRemark. We show more generally, that for any sequence \\( \\left\\{fjslqwer\\right\\} \\) with \\( fjslqwer \\rightarrow+\\infty \\) and \\( a_{qzxwvtnp+1}-fjslqwer \\rightarrow 0 \\), the set \\( nhaqtwes=\\left\\{fjslqwer-gnhsdapi: hjgrksla, qzxwvtnp \\geq 0\\right\\} \\) is dense in \\( \\mathbb{R} \\). Given \\( lbzgvhio \\geq 0 \\) and \\( aytksrnd>0 \\), fix \\( hjgrksla \\) such that \\( \\left|a_{tzbnwrce+1}-kzptynou\\right|<aytksrnd \\) for all \\( tzbnwrce \\geq hjgrksla \\). If \\( qzxwvtnp \\) is the smallest integer \\( \\geq hjgrksla \\) with \\( fjslqwer \\geq gnhsdapi+lbzgvhio \\), then \\( fjslqwer<gnhsdapi+lbzgvhio+aytksrnd \\), so \\( fjslqwer-gnhsdapi \\) is within \\( aytksrnd \\) of \\( lbzgvhio \\). This shows that \\( nhaqtwes \\) is dense in \\( [0, \\infty) \\), and by symmetry \\( nhaqtwes \\) is dense also in \\( (-\\infty, 0] \\).\n\nRemark. Let \\( ojxlerpu(vldqmpre) \\) be a function such that \\( ojxlerpu(vldqmpre) \\rightarrow+\\infty \\) and \\( ojxlerpu^{\\prime}(vldqmpre) \\rightarrow 0 \\) as \\( vldqmpre \\rightarrow+\\infty \\). The Mean Value Theorem shows that the hypotheses of the previous remark are satisfied by the sequence \\( fjslqwer=ojxlerpu(qzxwvtnp) \\).\n\nSolution 2. Fix \\( lbzgvhio \\in \\mathbb{R} \\) and \\( aytksrnd>0 \\). Then for sufficiently large positive integers \\( qzxwvtnp \\),\n\\[\n(qzxwvtnp+lbzgvhio)^{3}-(qzxwvtnp+lbzgvhio-aytksrnd)^{3}=3 qzxwvtnp^{2} aytksrnd+O(qzxwvtnp)>1\n\\]\nso \\( (qzxwvtnp+lbzgvhio-aytksrnd)^{3} \\leq\\left\\lfloor(qzxwvtnp+lbzgvhio)^{3}\\right\\rfloor \\leq(qzxwvtnp+lbzgvhio)^{3} \\), and \\( \\sqrt[3]{\\left\\lfloor(qzxwvtnp+lbzgvhio)^{3}\\right\\rfloor} \\) is within \\( aytksrnd \\) of \\( qzxwvtnp+lbzgvhio \\). Hence\n\\[\n\\lim _{qzxwvtnp \\rightarrow \\infty}\\left(\\sqrt[3]{\\left\\lfloor(qzxwvtnp+lbzgvhio)^{3}\\right\\rfloor}-\\sqrt[3]{qzxwvtnp}\\right)=lbzgvhio\n\\]\n\nSolution 3. As in Solution \\( 1, \\sqrt[3]{qzxwvtnp+1}-\\sqrt[3]{qzxwvtnp} \\rightarrow 0 \\) as \\( qzxwvtnp \\rightarrow \\infty \\), so the set \\( nhaqtwes=\\{\\sqrt[3]{qzxwvtnp}-\\sqrt[3]{hjgrksla}\\} \\) contains arbitrarily small positive numbers. Also \\( nhaqtwes \\) is closed under multiplication by positive integers \\( psqdlmva \\) since \\( psqdlmva(\\sqrt[3]{qzxwvtnp}-\\sqrt[3]{hjgrksla})=\\sqrt[3]{psqdlmva^{3} qzxwvtnp}-\\sqrt[3]{psqdlmva^{3} hjgrksla} \\). Any set with the preceding two properties is dense in \\( [0, \\infty) \\), because any finite open interval \\( (a, b) \\) in \\( [0, \\infty) \\) contains a multiple of any element of \\( nhaqtwes \\cap(0, b-a) \\). By symmetry \\( nhaqtwes \\) is dense in \\( (-\\infty, 0] \\) too.\n\nSolution 4. Let \\( wmpqzeru=(qzxwvtnp \\sqrt[3]{5} \\bmod 1) \\in[0,1] \\). As in the remark following 1988B3, \\( \\left\\{wmpqzeru\\right\\} \\) is dense in \\( [0,1] \\). Thus given \\( aytksrnd>0 \\), we can find \\( qzxwvtnp \\) such that \\( qzxwvtnp \\sqrt[3]{5}-lbzgvhio \\) is within \\( aytksrnd \\) of some integer \\( hjgrksla \\geq 0 \\). Then \\( \\sqrt[3]{5 qzxwvtnp^{3}}-\\sqrt[3]{hjgrksla^{3}}=qzxwvtnp \\sqrt[3]{5}-hjgrksla \\) is within \\( aytksrnd \\) of \\( lbzgvhio \\)."
    },
    "kernel_variant": {
      "question": "Let $k\\ge 1$ and let $d_{1},\\dots ,d_{k}\\ge 1$ be fixed.  \n\nFix a step vector  \n\\[\nq=(q_{1},\\dots ,q_{k})\\in\\mathbb{N}^{k},\\qquad q_{i}\\ge 1\\;(1\\le i\\le k),\n\\]\nand parameters  \n\\[\n0<\\alpha_{ij}<1,\\qquad c_{ij}\\in\\mathbb{R}\\setminus\\{0\\},\n\\qquad(i,j)\\in\\{1,\\dots ,k\\}\\times\\{1,\\dots ,d_{i}\\}.\n\\]\n\nDefine  \n\\[\nF(n_{1},\\dots ,n_{k})\n   =\\sum_{i=1}^{k}\\sum_{j=1}^{d_{i}}\n        c_{ij}\\,n_{i}^{\\alpha_{ij}},\n        \\qquad (n_{1},\\dots ,n_{k})\\in\\mathbb{N}^{k},\n\\]\nand put  \n\\[\n\\alpha_{*}:=\\max_{1\\le i\\le k,\\,1\\le j\\le d_{i}}\\alpha_{ij}\\quad(<1).\n\\]\n\nFor every coordinate $i$ set  \n\\[\n\\gamma_{i}:=\\sum_{j:\\,\\alpha_{ij}=\\alpha_{*}}c_{ij}.\n\\]\n\nNon–degeneracy hypothesis  \n\\[\n\\text{\\rm (ND)}\\qquad F \\text{ is \\emph{not} constant }\n               \\text{ (equivalently, }\\Delta_{q}\\not\\equiv 0).\n\\]\nA convenient sufficient condition for (ND) is  \n$\\gamma_{i}\\neq 0$ for at least one index $i$.\n\nFor $n\\in\\mathbb{N}^{k}$ define the one–step difference  \n\\[\n\\Delta_{q}(n):=F(n+q)-F(n),\\qquad \nT_{q}:=\\Bigl\\{\\sum_{\\ell=1}^{r} z_{\\ell}\\,\\Delta_{q}(n^{(\\ell)}):\n           r\\ge 1,\\;z_{\\ell}\\in\\mathbb{Z},\\;n^{(\\ell)}\\in\\mathbb{N}^{k}\\Bigr\\}.\n\\]\nFor $K\\in\\mathbb{N}$ and $v\\in\\mathbb{N}^{k}$ put  \n\\[\n\\Delta_{q}^{(K)}(v):=F(v+Kq)-F(v),\\qquad\nS:=\\{\\Delta_{q}^{(K)}(v):v\\in\\mathbb{N}^{k},\\,K\\in\\mathbb{N}\\}.\n\\]\n\nProve the following statements.\n\n(a) (Vanishing one–step difference)  \nIf $\\min_{1\\le i\\le k}n_{i}\\to\\infty$, then $\\Delta_{q}(n)\\to 0$.\n\n(b) Assume (ND).  Then the additive subgroup $T_{q}$ is dense in $\\mathbb{R}$.\n\n(c) Multi–step structure.  \n(c1) $\\Delta_{q}^{(K)}(v)\\in T_{q}$ for all $v\\in\\mathbb{N}^{k}$ and $K\\in\\mathbb{N}$.  \n(c2) If every $c_{ij}$ has the same sign, then every element of $S$ has this sign; hence $S$ is contained in a half–line and is not dense.  \n(c3) Suppose there exist \\emph{two} coordinates $i_{+},i_{-}$ such that  \n\\[\n\\gamma_{i_{+}}>0,\\qquad \\gamma_{i_{-}}<0.\n\\]\nThen $S$ contains arbitrarily small positive and arbitrarily small negative numbers.\n\n(d) Show that hypothesis (c3) is indispensable: construct an explicit system with mixed signs but $\\gamma_{i}$ of one fixed sign for \\emph{all} $i$ and prove that in this example $S$ is contained in a half–line.  \n(Hint: $k=1,\\;q=1,\\;d_{1}=2,\\;\\alpha_{11}=0.2,\\;\\alpha_{12}=0.9,\\;c_{11}=1,\\;c_{12}=-1$.)\n\n(e) Give an explicit example in which $c_{ij}>0$ for all $(i,j)$ and verify that $S$ is not dense even though parts (a) and (b) hold.  \n(Example: $k=1,\\;q=1,\\;d_{1}=1,\\;\\alpha_{11}=1/2,\\;c_{11}=1$.)\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Throughout $C,C',\\dots$ denote positive constants depending only on the fixed data; their value may change from line to line.  Put  \n\\[\nm(n):=\\min_{1\\le i\\le k}n_{i},\\qquad \n\\alpha_{*}:=\\max_{i,j}\\alpha_{ij}\\quad(<1).\n\\]\n\n--------------------------------------------------------------------\n(a)  One–step differences vanish when every coordinate diverges\n--------------------------------------------------------------------\nFix $(i,j)$ and set $g(x)=x^{\\alpha_{ij}}$ for $x>0$.  Because $0<\\alpha_{ij}<1$ we have\n\\[\ng'(x)=\\alpha_{ij}x^{\\alpha_{ij}-1}\\xrightarrow[x\\to\\infty]{}0 .\n\\]\nBy the mean–value theorem,\n\\[\n(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\n  =g'(\\xi_{ij}(n))\\,q_{i},\\qquad n_{i}\\le\\xi_{ij}(n)\\le n_{i}+q_{i},\n\\]\nand monotonicity of $g'$ gives\n\\[\n\\lvert(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\\rvert\n   \\le \\alpha_{ij}q_{i}\\,n_{i}^{\\alpha_{ij}-1}.\n\\]\nHence\n\\[\n\\lvert\\Delta_{q}(n)\\rvert\n  \\le C\\sum_{i=1}^{k}n_{i}^{\\alpha_{*}-1}\n  \\le C'\\,m(n)^{\\alpha_{*}-1}\\xrightarrow[n\\to\\infty]{}0\n  \\qquad(\\alpha_{*}-1<0).\n\\]\n\n--------------------------------------------------------------------\n(b)  Density of $T_{q}$ under (ND)\n--------------------------------------------------------------------\nWe divide the argument into three steps.  The new Step 2 repairs the gap\npointed out in the review.\n\nStep 1.  A non–zero difference exists.  \nBecause of (ND) choose $n^{(0)}\\in\\mathbb{N}^{k}$ with  \n$\\Delta_{q}(n^{(0)})\\neq 0$.  Thus $T_{q}\\neq\\{0\\}$.\n\nStep 2.  Construction of infinitely many \\emph{small but non–zero} one–step\ndifferences.\n\nList the distinct exponents that occur in $F$ in decreasing order\n\\[\n\\alpha^{(1)}=\\alpha_{*}>\\alpha^{(2)}>\\dots>\\alpha^{(s)}>0 .\n\\]\nFor $r=1,\\dots ,s$ put\n\\[\n\\Gamma_{i}^{(r)}:=\\sum_{j:\\,\\alpha_{ij}=\\alpha^{(r)}}c_{ij},\n\\qquad\n\\Theta^{(r)}(w):=\\alpha^{(r)}\\sum_{i=1}^{k}q_{i}\\Gamma_{i}^{(r)}w_{i}^{\\alpha^{(r)}-1},\n\\]\nwhere $w=(w_{1},\\dots ,w_{k})$ is any vector with positive coordinates.\n\nBecause (ND) holds there exist $r_{0}$ and an index $i$ such that\n$\\Gamma_{i}^{(r_{0})}\\neq 0$.  Fix this $r_{0}$.  Choose integers\n\\[\nw_{i}=L^{\\,i-1}\\quad(1\\le i\\le k)\n\\]\nwith $L\\in\\mathbb{N}$ to be sent to $\\infty$ later.  Then\n\\[\n\\Theta^{(r_{0})}(w)\n   =\\alpha^{(r_{0})}\n     \\sum_{i=1}^{k}q_{i}\\Gamma_{i}^{(r_{0})}\n                 L^{(i-1)(\\alpha^{(r_{0})}-1)}.\n\\]\nWrite $b_{i}:=(i-1)(\\alpha^{(r_{0})}-1)$; because $\\alpha^{(r_{0})}-1<0$\nthe sequence $(b_{i})_{1\\le i\\le k}$ is strictly \\emph{decreasing}:\n$b_{1}=0>b_{2}>\\dots>b_{k}$.\n\nLet\n\\[\nI:=\\{i:\\Gamma_{i}^{(r_{0})}\\neq 0\\},\\qquad \ni_{0}:=\\mathop{\\arg\\max}_{i\\in I}b_{i}\\;(=\\min I).\n\\]\nHence $b_{i_{0}}=\\max_{i\\in I}b_{i}\\;(=0$ if $\\Gamma_{1}^{(r_{0})}\\neq 0)$.\nSet\n\\[\na_{0}:=\\alpha^{(r_{0})}q_{i_{0}}\\Gamma_{i_{0}}^{(r_{0})}\\neq 0,\n\\qquad \n\\Phi(L):=L^{-b_{i_{0}}}\\Theta^{(r_{0})}(w)\n       =a_{0}+\\sum_{i\\neq i_{0}}a_{i}\\,L^{\\,b_{i}-b_{i_{0}}}.\n\\]\nNow $b_{i}-b_{i_{0}}<0$ for $i\\neq i_{0}$, so\n\\[\n\\lim_{L\\to\\infty}\\Phi(L)=a_{0}\\neq 0.\n\\]\nConsequently there exists $L_{0}$ such that\n$\\Phi(L)$ has the fixed sign $\\operatorname{sgn}(a_{0})$ and is therefore\nnon–zero for all $L\\ge L_{0}$.  Because\n$L^{-b_{i_{0}}}>0$, the same conclusion holds for\n$\\Theta^{(r_{0})}(w)$:  \n\\[\n\\Theta^{(r_{0})}(w)\\neq 0\\quad\\text{for all integers }L\\ge L_{0}.\n\\]\n\nFix one such large $L$ and define the ray\n\\[\nn^{(t)}:=(tw_{1},\\dots ,tw_{k}),\\qquad t\\in\\mathbb{N},\\;t\\ge 1.\n\\]\n\nA straight–forward binomial expansion gives the asymptotic\n\\[\n\\Delta_{q}(n^{(t)})\n  =\\Theta^{(r_{0})}(w)\\,t^{\\alpha^{(r_{0})}-1}\n     +O\\!\\bigl(t^{\\alpha^{(r_{0})}-2}\\bigr)\n     +\\text{(smaller-order terms)},\n\\qquad t\\to\\infty .\n\\]\nBecause $\\Theta^{(r_{0})}(w)\\neq 0$ the right–hand side has the\n\\emph{fixed, non–zero} sign $\\operatorname{sgn}\\bigl(\\Theta^{(r_{0})}(w)\\bigr)$\nfor all sufficiently large $t$, and\n\\[\n\\lvert\\Delta_{q}(n^{(t)})\\rvert\\asymp t^{\\alpha^{(r_{0})}-1}\\xrightarrow[t\\to\\infty]{}0\n\\qquad(\\alpha^{(r_{0})}-1<0).\n\\]\nThus $T_{q}$ contains non–zero elements of arbitrarily small\nmagnitude.\n\nStep 3.  Closed additive subgroups of $\\mathbb{R}$.  \nA closed additive subgroup of $\\mathbb{R}$ is either $\\{0\\}$, of the\nform $a\\mathbb{Z}$ with $a>0$, or dense; see, for instance,\nRudin, \\emph{Real and Complex Analysis}, Lemma 1.22.  \nBecause $T_{q}$ possesses non–zero elements of arbitrarily small absolute value,\nthe first two alternatives are impossible; hence $\\overline{T_{q}}=\\mathbb{R}$\nand $T_{q}$ is dense.\n\n--------------------------------------------------------------------\n(c1)  Multi–step differences lie in $T_{q}$\n--------------------------------------------------------------------\n\\[\n\\Delta_{q}^{(K)}(v)\n  =\\sum_{t=0}^{K-1}\\Bigl[F\\bigl(v+(t+1)q\\bigr)-F\\bigl(v+tq\\bigr)\\Bigr]\n  =\\sum_{t=0}^{K-1}\\Delta_{q}\\bigl(v+tq\\bigr)\\in T_{q}.\n\\]\n\n--------------------------------------------------------------------\n(c2)  All coefficients have the same sign\n--------------------------------------------------------------------\nWrite $c_{ij}=\\sigma\\lvert c_{ij}\\rvert$ with $\\sigma\\in\\{+1,-1\\}$.\nSince $(n_{i}+q_{i})^{\\alpha_{ij}}>n_{i}^{\\alpha_{ij}}$,\n\\[\n\\sigma\\cdot\\Delta_{q}(n)\n  =\\sum_{i,j}\\lvert c_{ij}\\rvert\n     \\bigl[(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\\bigr]\n  >0 ,\n\\]\nso $\\sigma\\cdot S\\subset(0,\\infty)$.\nTherefore $S$ lies in a half–line and cannot be dense.\n\n--------------------------------------------------------------------\n(c3)  Opposite signs for the leading exponent\n--------------------------------------------------------------------\nAssume $\\gamma_{i_{+}}>0$ and $\\gamma_{i_{-}}<0$.  \nFix $\\beta>1$ and for $N\\in\\mathbb{N}$ define  \n\\[\nn^{(+)}_{i}(N)=\n\\begin{cases}\nN,& i=i_{+},\\\\[4pt]\n\\lceil N^{\\beta}\\rceil,& i\\neq i_{+},\n\\end{cases}\n\\qquad\nn^{(-)}_{i}(N)=\n\\begin{cases}\nN,& i=i_{-},\\\\[4pt]\n\\lceil N^{\\beta}\\rceil,& i\\neq i_{-}.\n\\end{cases}\n\\]\nExpanding as before gives\n\\[\n\\Delta_{q}\\bigl(n^{(+)}(N)\\bigr)\n  =\\alpha_{*}q_{i_{+}}\\gamma_{i_{+}}\\,N^{\\alpha_{*}-1}\n   +O\\!\\bigl(N^{\\alpha_{*}-2}\\bigr)\n   +O\\!\\bigl(N^{\\beta(\\alpha_{*}-1)}\\bigr).\n\\]\nBecause $\\alpha_{*}-1<0$ and $\\beta>1$, the last error term is\n$o\\!\\bigl(N^{\\alpha_{*}-1}\\bigr)$.  \nConsequently $\\Delta_{q}\\bigl(n^{(+)}(N)\\bigr)$ is positive for all large\n$N$ and tends to $0$; a similar argument with $n^{(-)}(N)$ produces\nsmall negative values.  Hence $S$ contains arbitrarily small numbers of\nboth signs.\n\n--------------------------------------------------------------------\n(d)  Necessity of the sign–balance hypothesis\n--------------------------------------------------------------------\nTake\n\\[\nk=1,\\;q=1,\\;d_{1}=2,\\;\n\\alpha_{11}=0.2,\\;\\alpha_{12}=0.9=\\alpha_{*},\\;\nc_{11}=1,\\;c_{12}=-1.\n\\]\nHere $\\gamma_{1}=c_{12}=-1<0$, so the hypothesis of (c3) fails.\n\nFor $n\\ge 0$,\n\\[\n\\Delta_{1}(n)\n   =-(n+1)^{0.9}+n^{0.9}+(n+1)^{0.2}-n^{0.2}\n   =-\\varphi_{0.9}(n)+\\varphi_{0.2}(n),\n\\]\nwhere  \n\\[\n\\varphi_{\\beta}(x):=(x+1)^{\\beta}-x^{\\beta},\\qquad 0<\\beta<1.\n\\]\n\nClaim: $\\varphi_{0.9}(x)>\\varphi_{0.2}(x)$ for every $x>0$.  \n\nProof of the claim.  \nFor fixed $x>0$ define $h(\\beta)=(x+1)^{\\beta}-x^{\\beta}$ on $(0,1)$.  \nThen\n\\[\nh'(\\beta)=(x+1)^{\\beta}\\ln(x+1)-x^{\\beta}\\ln x>0 ,\n\\]\nbecause $(x+1)^{\\beta}>x^{\\beta}$ and $\\ln(x+1)>\\ln x$.  \nThus $h$ is strictly increasing in $\\beta$, and the claim follows.\n\nTherefore $\\Delta_{1}(n)<0$ for all $n\\ge 0$.  Multi–step differences are sums of such negative numbers, so  \n\\[\nS\\subset(-\\infty,0] .\n\\]\nMixed signs of the $c_{ij}$ alone are insufficient; the balance condition in (c3) is indeed necessary.\n\n--------------------------------------------------------------------\n(e)  All coefficients positive – $S$ still not dense\n--------------------------------------------------------------------\nExample: $k=1$, $q=1$, $d_{1}=1$, $\\alpha_{11}=1/2$, $c_{11}=1$.  \nThen $F(n)=n^{1/2}$ and\n\\[\n\\Delta_{1}(n)=\\sqrt{n+1}-\\sqrt{n}>0\\qquad(n\\ge 0),\n\\]\nhence $S\\subset(0,\\infty)$, so $S$ is not dense even though parts (a)\nand (b) hold.  This confirms (c2).\n\n\\hfill$\\square$\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.712633",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality:  The variable n now lives in ℕᵏ with k ≥ 2,\n   so estimates involve several coordinates and multivariate stepping.\n\n2. Additional structural constraints:  \n   • Differences are taken only in a fixed direction q,  \n   • all exponents {α_{ij}} must be simultaneously ℚ–independent,  \n   • density must be shown not merely for the set of single\n     differences S_q but for its entire ℤ–span T_q,\n   • finally the full difference set S is considered under the\n     congruence restriction u≡v (mod q).\n\n3. Deeper theory required:  \n   • Multivariate Mean Value Theorem (to handle part (a)),  \n   • Additive subgroup arguments together with Diophantine\n     approximation (part (b)),  \n   • A telescoping decomposition in k dimensions to bridge\n     S_q and S (part (c)).\n\n4. Multiple interacting ideas:  One needs analysis (derivative\n   estimates), algebra (ℤ–spans and additive subgroups), and discrete\n   geometry (lattice walks consistent with the modulus q).\n\nAltogether these elements make the enhanced variant significantly more\ntechnical and conceptually richer than both the original single-index\nproblem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $k\\ge 1$ and let $d_{1},\\dots ,d_{k}\\ge 1$.  \n\nFix a step vector  \n\\[\nq=(q_{1},\\dots ,q_{k})\\in\\mathbb{N}^{k},\\qquad q_{i}\\ge 1\\;(1\\le i\\le k),\n\\]\nand parameters  \n\\[\n0<\\alpha_{ij}<1,\\qquad c_{ij}\\in\\mathbb{R}\\setminus\\{0\\},\n\\qquad(i,j)\\in\\{1,\\dots ,k\\}\\times\\{1,\\dots ,d_{i}\\}.\n\\]\n\nDefine  \n\\[\nF(n_{1},\\dots ,n_{k})\n   =\\sum_{i=1}^{k}\\sum_{j=1}^{d_{i}}\n        c_{ij}\\,n_{i}^{\\alpha_{ij}},\n        \\qquad (n_{1},\\dots ,n_{k})\\in\\mathbb{N}^{k},\n\\]\nand put  \n\\[\n\\alpha_{*}:=\\max_{1\\le i\\le k,\\,1\\le j\\le d_{i}}\\alpha_{ij}\\quad(<1).\n\\]\n\nFor every coordinate $i$ set  \n\\[\n\\gamma_{i}:=\\sum_{j:\\,\\alpha_{ij}=\\alpha_{*}}c_{ij}.\n\\]\n\nNon–degeneracy hypothesis  \n\\[\n\\text{\\rm (ND)}\\qquad F \\text{ is \\emph{not} constant }\n               \\text{ (equivalently, }\\Delta_{q}\\not\\equiv 0).\n\\]\nA convenient sufficient condition for (ND) is  \n$\\gamma_{i}\\neq 0$ for at least one index $i$.\n\nFor $n\\in\\mathbb{N}^{k}$ define the one–step difference  \n\\[\n\\Delta_{q}(n):=F(n+q)-F(n),\\qquad \nT_{q}:=\\Bigl\\{\\sum_{\\ell=1}^{r} z_{\\ell}\\,\\Delta_{q}(n^{(\\ell)}):\n           r\\ge 1,\\;z_{\\ell}\\in\\mathbb{Z},\\;n^{(\\ell)}\\in\\mathbb{N}^{k}\\Bigr\\}.\n\\]\nFor $K\\in\\mathbb{N}$ and $v\\in\\mathbb{N}^{k}$ put  \n\\[\n\\Delta_{q}^{(K)}(v):=F(v+Kq)-F(v),\\qquad\nS:=\\{\\Delta_{q}^{(K)}(v):v\\in\\mathbb{N}^{k},\\,K\\in\\mathbb{N}\\}.\n\\]\n\nProve the following statements.\n\n(a) (Vanishing one–step difference)  \nIf $\\min_{1\\le i\\le k}n_{i}\\to\\infty$, then $\\Delta_{q}(n)\\to 0$.\n\n(b) Assume (ND).  Then the additive subgroup $T_{q}$ is dense in $\\mathbb{R}$.\n\n(c) Multi–step structure.  \n(c1) $\\Delta_{q}^{(K)}(v)\\in T_{q}$ for all $v\\in\\mathbb{N}^{k}$ and $K\\in\\mathbb{N}$.  \n(c2) If every $c_{ij}$ has the same sign, then every element of $S$ has this sign; hence $S$ is contained in a half–line and is not dense.  \n(c3) Suppose there exist \\emph{two} coordinates $i_{+},i_{-}$ such that  \n\\[\n\\gamma_{i_{+}}>0,\\qquad \\gamma_{i_{-}}<0.\n\\]\nThen $S$ contains arbitrarily small positive and arbitrarily small negative numbers.\n\n(d) Show that hypothesis (c3) is indispensable: construct an explicit system with mixed signs but $\\gamma_{i}$ of one fixed sign for \\emph{all} $i$ and prove that in this example $S$ is contained in a half–line.  \n(Hint: $k=1,\\;q=1,\\;d_{1}=2,\\;\\alpha_{11}=0.2,\\;\\alpha_{12}=0.9,\\;c_{11}=1,\\;c_{12}=-1$.)\n\n(e) Give an explicit example in which $c_{ij}>0$ for all $(i,j)$ and verify that $S$ is not dense even though parts (a) and (b) hold.  \n(Example: $k=1,\\;q=1,\\;d_{1}=1,\\;\\alpha_{11}=1/2,\\;c_{11}=1$.)",
      "solution": "Throughout $C,C',\\dots$ denote positive constants depending only on the fixed data; their value may change from line to line.  Put  \n\\[\nm(n):=\\min_{1\\le i\\le k}n_{i},\\qquad \n\\alpha_{*}:=\\max_{i,j}\\alpha_{ij}\\quad(<1).\n\\]\n\n------------------------------------------------------------\n(a)  One–step differences vanish when every coordinate diverges\n------------------------------------------------------------\nFix $(i,j)$ and set $g(x)=x^{\\alpha_{ij}}$ for $x>0$.  Because $0<\\alpha_{ij}<1$ we have\n\\[\ng'(x)=\\alpha_{ij}x^{\\alpha_{ij}-1}\\xrightarrow[x\\to\\infty]{}0 .\n\\]\nBy the mean–value theorem,\n\\[\n(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\n  =g'(\\xi_{ij}(n))\\,q_{i},\\qquad n_{i}\\le\\xi_{ij}(n)\\le n_{i}+q_{i},\n\\]\nand monotonicity of $g'$ gives\n\\[\n\\lvert(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\\rvert\n   \\le \\alpha_{ij}q_{i}\\,n_{i}^{\\alpha_{ij}-1}.\n\\]\nHence\n\\[\n\\lvert\\Delta_{q}(n)\\rvert\n  \\le C\\sum_{i=1}^{k}n_{i}^{\\alpha_{*}-1}\n  \\le C'\\,m(n)^{\\alpha_{*}-1}\\xrightarrow[n\\to\\infty]{}0\n  \\qquad(\\alpha_{*}-1<0).\n\\]\n\n------------------------------------------------------------\n(b)  Density of $T_{q}$ under (ND)\n------------------------------------------------------------\nWe divide the argument into three steps, inserting in Step 2 the missing justification demanded by the reviewers.\n\nStep 1.  A non–zero difference exists.  \nBecause of (ND) choose $n^{(0)}\\in\\mathbb{N}^{k}$ with  \n$\\Delta_{q}(n^{(0)})\\neq 0$.  Thus $T_{q}\\neq\\{0\\}$.\n\nStep 2.  Construction of infinitely many \\emph{small but non–zero} one–step\ndifferences.\n\nList the distinct exponents that occur in $F$ in decreasing order\n\\[\n\\alpha^{(1)}=\\alpha_{*}>\\alpha^{(2)}>\\dots>\\alpha^{(s)}>0 .\n\\]\nFor $r=1,\\dots ,s$ put\n\\[\n\\Gamma_{i}^{(r)}:=\\sum_{j:\\,\\alpha_{ij}=\\alpha^{(r)}}c_{ij},\n\\qquad\n\\Theta^{(r)}(w):=\\alpha^{(r)}\\sum_{i=1}^{k}q_{i}\\Gamma_{i}^{(r)}w_{i}^{\\alpha^{(r)}-1},\n\\]\nwhere $w=(w_{1},\\dots ,w_{k})$ is any vector with positive coordinates.\n\nBecause $F$ is not constant there exists an index\n$r_{0}\\in\\{1,\\dots ,s\\}$ and a coordinate $i_{0}$ such that\n$\\Gamma_{i_{0}}^{(r_{0})}\\neq 0$.\nChoose integers\n\\[\nw_{i}=L^{\\,i-1}\\quad(1\\le i\\le k)\n\\]\nwith $L$ an integer parameter to be fixed later.  Then\n\\[\n\\Theta^{(r_{0})}(w)\n   =\\alpha^{(r_{0})}\n     \\sum_{i=1}^{k}q_{i}\\Gamma_{i}^{(r_{0})}\n                 L^{(i-1)(\\alpha^{(r_{0})}-1)}.\n\\]\nWrite $b_{i}:=(i-1)(\\alpha^{(r_{0})}-1)$; these real exponents are\n\\emph{strictly decreasing} because $\\alpha^{(r_{0})}-1<0$.  Let  \n$b_{\\min}:=\\min_{1\\le i\\le k}b_{i}=b_{i_{0}}$ and set  \n\\[\n\\Phi(L):=L^{-b_{\\min}}\\Theta^{(r_{0})}(w)\n       =a_{0}+\\sum_{i\\neq i_{0}}a_{i}\\,L^{b_{i}-b_{\\min}},\n\\qquad a_{0}:=\\alpha^{(r_{0})}q_{i_{0}}\\Gamma_{i_{0}}^{(r_{0})}\\neq 0 .\n\\]\nAll other exponents $b_{i}-b_{\\min}$ are \\emph{positive}.\nConsequently\n\\[\n\\lim_{L\\to\\infty}\\Phi(L)=a_{0}\\neq 0\n\\quad\\Longrightarrow\\quad\n\\lim_{L\\to\\infty}\\Theta^{(r_{0})}(w)L^{-b_{\\min}}=a_{0}\\neq 0 .\n\\]\nThus $\\Theta^{(r_{0})}(w)$ assumes the fixed non–zero sign\n$\\operatorname{sgn}(a_{0})$ for \\emph{all sufficiently large integers $L$}.\nIn particular, $\\Theta^{(r_{0})}(w)\\neq 0$ for every such $L$; hence the set of\nintegers $L$ with $\\Theta^{(r_{0})}(w)=0$ is finite.\n\nFix one large $L$ satisfying $\\Theta^{(r_{0})}(w)\\neq 0$ and define the ray\n\\[\nn^{(t)}:=(tw_{1},\\dots ,tw_{k}),\\qquad t\\in\\mathbb{N},\\;t\\ge 1.\n\\]\n\nA straight–forward binomial expansion gives the asymptotic\n\\[\n\\Delta_{q}(n^{(t)})\n  =\\Theta^{(r_{0})}(w)\\,t^{\\alpha^{(r_{0})}-1}\n     +O\\!\\bigl(t^{\\alpha^{(r_{0})}-2}\\bigr)\n     +\\text{(smaller-order terms)},\n\\qquad t\\to\\infty .\n\\]\nBecause $\\Theta^{(r_{0})}(w)\\neq 0$ the right–hand side has the\n\\emph{fixed, non–zero} sign $\\operatorname{sgn}\\bigl(\\Theta^{(r_{0})}(w)\\bigr)$\nfor all sufficiently large $t$, and\n\\[\n\\lvert\\Delta_{q}(n^{(t)})\\rvert\\asymp t^{\\alpha^{(r_{0})}-1}\\xrightarrow[t\\to\\infty]{}0\n\\qquad(\\alpha^{(r_{0})}-1<0).\n\\]\nThus $T_{q}$ contains non–zero elements of arbitrarily small\nmagnitude, irrespective of the values of the $\\gamma_{i}$.\n\nStep 3.  Closed additive subgroups of $\\mathbb{R}$.  \nA closed additive subgroup of $\\mathbb{R}$ is either $\\{0\\}$, of the\nform $a\\mathbb{Z}$ with $a>0$, or dense; see, for instance,\nRudin, \\emph{Real and Complex Analysis}, Lemma 1.22.  \nBecause $T_{q}$ possesses non–zero elements of arbitrarily small absolute value,\nthe first two alternatives are impossible; hence $\\overline{T_{q}}=\\mathbb{R}$\nand $T_{q}$ is dense.\n\n------------------------------------------------------------\n(c1)  Multi–step differences lie in $T_{q}$\n------------------------------------------------------------\n\\[\n\\Delta_{q}^{(K)}(v)\n  =\\sum_{t=0}^{K-1}\\Bigl[F\\bigl(v+(t+1)q\\bigr)-F\\bigl(v+tq\\bigr)\\Bigr]\n  =\\sum_{t=0}^{K-1}\\Delta_{q}\\bigl(v+tq\\bigr)\\in T_{q}.\n\\]\n\n------------------------------------------------------------\n(c2)  All coefficients have the same sign\n------------------------------------------------------------\nWrite $c_{ij}=\\sigma\\lvert c_{ij}\\rvert$ with $\\sigma\\in\\{+1,-1\\}$.\nSince $(n_{i}+q_{i})^{\\alpha_{ij}}>n_{i}^{\\alpha_{ij}}$,\n\\[\n\\sigma\\cdot\\Delta_{q}(n)\n  =\\sum_{i,j}\\lvert c_{ij}\\rvert\n     \\bigl[(n_{i}+q_{i})^{\\alpha_{ij}}-n_{i}^{\\alpha_{ij}}\\bigr]\n  >0 ,\n\\]\nso $\\sigma\\cdot S\\subset(0,\\infty)$.\nTherefore $S$ lies in a half–line and cannot be dense.\n\n------------------------------------------------------------\n(c3)  Opposite signs for the leading exponent\n------------------------------------------------------------\nAssume $\\gamma_{i_{+}}>0$ and $\\gamma_{i_{-}}<0$.  \nFix $\\beta>1$ and for $N\\in\\mathbb{N}$ define  \n\\[\nn^{(+)}_{i}(N)=\n\\begin{cases}\nN,& i=i_{+},\\\\[4pt]\n\\lceil N^{\\beta}\\rceil,& i\\neq i_{+},\n\\end{cases}\n\\qquad\nn^{(-)}_{i}(N)=\n\\begin{cases}\nN,& i=i_{-},\\\\[4pt]\n\\lceil N^{\\beta}\\rceil,& i\\neq i_{-}.\n\\end{cases}\n\\]\nExpanding as before gives\n\\[\n\\Delta_{q}\\bigl(n^{(+)}(N)\\bigr)\n  =\\alpha_{*}q_{i_{+}}\\gamma_{i_{+}}\\,N^{\\alpha_{*}-1}\n   +O\\!\\bigl(N^{\\alpha_{*}-2}\\bigr)\n   +O\\!\\bigl(N^{\\beta(\\alpha_{*}-1)}\\bigr).\n\\]\nBecause $\\alpha_{*}-1<0$ and $\\beta>1$, the last error term is\n$o\\!\\bigl(N^{\\alpha_{*}-1}\\bigr)$.  \nConsequently $\\Delta_{q}\\bigl(n^{(+)}(N)\\bigr)$ is positive for all large\n$N$ and tends to $0$; a similar argument with $n^{(-)}(N)$ produces\nsmall negative values.  Hence $S$ contains arbitrarily small numbers of\nboth signs.\n\n------------------------------------------------------------\n(d)  Necessity of the sign–balance hypothesis\n------------------------------------------------------------\nTake\n\\[\nk=1,\\;q=1,\\;d_{1}=2,\\;\n\\alpha_{11}=0.2,\\;\\alpha_{12}=0.9=\\alpha_{*},\\;\nc_{11}=1,\\;c_{12}=-1.\n\\]\nHere $\\gamma_{1}=c_{12}=-1<0$, so the hypothesis of (c3) fails.\n\nFor $n\\ge 0$,\n\\[\n\\Delta_{1}(n)\n   =-(n+1)^{0.9}+n^{0.9}+(n+1)^{0.2}-n^{0.2}\n   =-\\varphi_{0.9}(n)+\\varphi_{0.2}(n),\n\\]\nwhere  \n\\[\n\\varphi_{\\beta}(x):=(x+1)^{\\beta}-x^{\\beta},\\qquad 0<\\beta<1.\n\\]\n\nClaim: $\\varphi_{0.9}(x)>\\varphi_{0.2}(x)$ for every $x>0$.  \n\nProof of the claim.  \nFor fixed $x>0$ define $h(\\beta)=(x+1)^{\\beta}-x^{\\beta}$ on $(0,1)$.  \nThen\n\\[\nh'(\\beta)=(x+1)^{\\beta}\\ln(x+1)-x^{\\beta}\\ln x>0 ,\n\\]\nbecause $(x+1)^{\\beta}>x^{\\beta}$ and $\\ln(x+1)>\\ln x$.  \nThus $h$ is strictly increasing in $\\beta$, and the claim follows.\n\nTherefore $\\Delta_{1}(n)<0$ for all $n\\ge 0$.  Multi–step differences are sums of such negative numbers, so  \n\\[\nS\\subset(-\\infty,0] .\n\\]\nMixed signs of the $c_{ij}$ alone are insufficient; the balance condition in (c3) is indeed necessary.\n\n------------------------------------------------------------\n(e)  All coefficients positive – $S$ still not dense\n------------------------------------------------------------\nExample: $k=1$, $q=1$, $d_{1}=1$, $\\alpha_{11}=1/2$, $c_{11}=1$.  \nThen $F(n)=n^{1/2}$ and\n\\[\n\\Delta_{1}(n)=\\sqrt{n+1}-\\sqrt{n}>0\\qquad(n\\ge 0),\n\\]\nhence $S\\subset(0,\\infty)$, so $S$ is not dense even though parts (a)\nand (b) hold.  This confirms (c2).\n\n\\hfill$\\square$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.555348",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensionality:  The variable n now lives in ℕᵏ with k ≥ 2,\n   so estimates involve several coordinates and multivariate stepping.\n\n2. Additional structural constraints:  \n   • Differences are taken only in a fixed direction q,  \n   • all exponents {α_{ij}} must be simultaneously ℚ–independent,  \n   • density must be shown not merely for the set of single\n     differences S_q but for its entire ℤ–span T_q,\n   • finally the full difference set S is considered under the\n     congruence restriction u≡v (mod q).\n\n3. Deeper theory required:  \n   • Multivariate Mean Value Theorem (to handle part (a)),  \n   • Additive subgroup arguments together with Diophantine\n     approximation (part (b)),  \n   • A telescoping decomposition in k dimensions to bridge\n     S_q and S (part (c)).\n\n4. Multiple interacting ideas:  One needs analysis (derivative\n   estimates), algebra (ℤ–spans and additive subgroups), and discrete\n   geometry (lattice walks consistent with the modulus q).\n\nAltogether these elements make the enhanced variant significantly more\ntechnical and conceptually richer than both the original single-index\nproblem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}