Arithmetic structures in smooth subsets of F_p

TL;DR

This paper establishes lower bounds on the sum over solutions to linear equations in Z_N for functions with small Fourier coefficients, extending previous work and using techniques related to Green's arithmetic regularity lemma.

Contribution

It generalizes earlier results by providing bounds for smooth functions on Z_N solving linear equations, employing Fourier analysis and regularity methods.

Findings

Bounded sums over solutions for smooth functions

Applicable to functions supported on sumsets and pseudoprimes

Extends previous results to a broader class of functions

Abstract

Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in Z_N to a_1 x_1 + ... + a_d x_d == 0 (mod N). Note that d = 3 and a_1 = a_2 = 1 and a_3 = -2 is the case where x_1,x_2,x_3 are in arithmetic progression. By ``smooth enough'' we mean that the sum of squares of the lower order Fourier coefficients of f is ``small'', a property shared by many naturally-occurring functions, among them certain ones supported on sumsets and on certain types of pseudoprimes. The paper can be thought of as a generalization of another result of the author, which dealt with a F_p^n analogue of the problem. It appears that the method in that paper, and to a more limited extent the present paper, uses ideas similar to those of B. Green's ``arithmetic regularity lemma'', as we explain in the paper.