Zero-sum subsequences in bounded-sum {-,}-sequences

Alec Sun

Abstract: We study the problem of finding zero-sum blocks in bounded-sum sequences, which was introduced by Caro, Hansberg, and Montejano. Caro et al. determine the minimum {−1,1}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">}</mo></math>-sequence length for when there exist k consecutive terms that sum to zero. We determine the corresponding minimum sequence length when the set {−1,1}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">}</mo></math> is replaced by {−r,s}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">s</mi><mo stretchy="false" is="true">}</mo></math> for arbitrary positive integers r and s. This confirms a conjecture of theirs. We also construct {−1,1}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">}</mo></math>-sequences of length quadratic in k that avoid k terms indexed by an arithmetic progression that sum to zero. This solves a second conjecture of theirs in the case of {−1,1}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">}</mo></math>-sequences on zero-sum arithmetic subsequences. Finally, we give for sufficiently large k a superlinear lower bound on the minimum sequence length to find a zero-sum arithmetic progression for general {−r,s}<math><mo stretchy="false" is="true">{</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">s</mi><mo stretchy="false" is="true">}</mo></math>-sequences.