Positive exponential sums and odd polynomials

TL;DR

The paper constructs special trigonometric polynomials related to odd polynomial values and uses them to analyze the maximal size of sets avoiding differences in those polynomial values, providing new proofs and interpretations.

Contribution

It introduces a novel construction of non-negative trigonometric polynomials with spectra in polynomial value sets, offering an alternative proof for bounds on sets avoiding polynomial differences.

Findings

Constructed trigonometric polynomials with spectrum in polynomial value sets

Provided an alternative proof for the maximal size of difference-avoiding sets

Discussed ergodic interpretations of the bounds

Abstract

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient a_{0}=O((\log n)^{-1/k}). This gives an alternative proof for the maximal possible cardinality of a set A, so that A-A does not contain an element of f(x). We also discuss other interpretations and an ergodic characterization of that bound.