On a paper of Erd\"os and Szekeres

TL;DR

This paper extends Erdös and Szekeres' work on the behavior of specific products involving integer sequences, providing new propositions that deepen understanding of their properties in relation to proportional subsets and sets with large arithmetic diameter.

Contribution

It introduces new propositions that advance the analysis of products of the form \\prod_{j=1}^n (1 - z^{a_j}) in the context of Erdös and Szekeres' problems, focusing on particular integer sets.

Findings

Propositions 1.1 -- 1.3 offer new insights into product behavior.

Results relate to proportional subsets of \\{1, ..., n\\}.

Results also address sets with large arithmetic diameter.

Abstract

Propositions 1.1 -- 1.3 stated below contribute to results and certain problems considered in a paper by Erdos and Szekeres, on the behavior of products $\prod^n_1 (1-z^{a_j}), 1\leq a_1\leq \cdots\leq a_n$ integers. In the discussion, $\{a_1, \ldots, a_n \}$ will be either a proportional subset of $\{1, \ldots, n\}$ or a set of large arithmetic diameter.