Recurrence and non-uniformity of bracket polynomials

TL;DR

This paper demonstrates that for a broad class of bracket polynomials, the associated exponential functions have a non-trivial Gowers norm, indicating structured behavior, and provides elementary proofs for some cases.

Contribution

It establishes a uniform lower bound on Gowers norms for bracket polynomial exponential functions and offers elementary proofs in special cases, extending understanding of their recurrence properties.

Findings

Gowers U^k[N]-norms are bounded away from zero for certain bracket polynomial functions.

Values of bracket polynomials are close to zero on a positive proportion of points.

Elementary proofs are provided for specific low-complexity cases.

Abstract

A bracket polynomial on the integers is a function formed using the operations of addition, multiplication and taking fractional parts. For a fairly large class of bracket polynomials we show that if p is a bracket polynomial of degree k-1 on [N] then the function f defined by f(n) = e(p(n)) has Gowers U^k[N]-norm bounded away from zero, uniformly in N. We establish this result by first reducing it to a certain recurrence property of sets of bracket polynomials. Specifically, for a fairly large class of bracket polynomials we show that if p_1, ..., p_r are bracket polynomials then their values, modulo 1, are all close to zero on at least some constant proportion of the points 1, ..., N. The proofs rely on two deep results from the literature. The first is work of V. Bergelson and A. Leibman showing that an arbitrary bracket polynomial can be expressed in terms of a polynomial sequence on a nilmanifold. The second is a theorem of B. Green and T. Tao describing the quantitative distribution properties of such polynomial sequences. We give elementary alternative proofs of the first result, without reference to nilmanifolds, in certain "low-complexity" special cases.