On solution-free sets for simultaneous diagonal polynomials

TL;DR

This paper proves that sets with certain uniformity properties contain the expected solutions to a system of diagonal equations, and that solution-free sets must be very sparse, with density tending to zero as the range grows.

Contribution

It establishes a connection between Gowers' uniformity and the existence of solutions to diagonal polynomial systems, and provides bounds on the size of solution-free sets.

Findings

Sets with Gowers' uniformity contain expected solutions

Solution-free sets are extremely sparse, with density tending to zero

Provides quantitative bounds on the size of solution-free sets

Abstract

We consider a translation and dilation invariant system consisting of k diagonal equations of degrees 1,2,...,k with integer coefficients in s variables, where s is sufficiently large in terms of k. We show via the Hardy-Littlewood circle method that if a subset A of the natural numbers restricted to the interval [1,N] satisfies Gowers' definition of uniformity of degree k, then it furnishes roughly the expected number of simultaneous solutions to the given equations. If A furnishes no non-trivial solutions to the given system, then we show that the number of elements of A in [1,N] grows no faster than a constant multiple of N/(log log N)^{-c} as N grows to infinity, where c>0 is a constant dependent only on k. In particular, we show that the density of A in [1,N] tends to 0 as N tends to infinity.