2007/08/27 by Ernie Croot, Croot, Ernie
Mathematics · #11P99 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.CO #math.NT #msc:11P99
paper · pdf · doi:10.48550/arxiv.0708.3689
This is a very preliminary draft. Future drafts will have cleaner proofs and tighter notation
arxiv created 2007/08/27 · openalex publication_date 2007/08/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix integers a1,...,ad satisfying a1 + ... + ad = 0. Suppose that f : ZN -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x1)...f(xd) over all solutions (x1,...,xd) in ZN to a1 x1 + ... + ad xd == 0 (mod N). Note that d = 3 and a1 = a2 = 1 and a3 = -2 is the case where x1,x2,x3 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 Fpn 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.