Lower bounds in the polynomial Szemerédi theorem

Abstract

We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's construction for sets lacking a square difference. We also extend Ruzsa's construction to sets lacking polynomial differences for a wide class of univariate polynomials. Finally, we turn to multivariate differences, constructing a set of order $N^{1/2}$ lacking a difference equal to a sum of two squares. This is in contrast to the analogous problem of sets lacking a difference equal to a prime minus one, where the current record is of order $N^{o(1)}.$

11 pages

References in corpus (3)

• Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case
• On a theorem of Sárközy for difference sets and shifted primes
• Square-Difference-Free Sets of Size Omega(n^{0.7334...})

Cited by in corpus (1)

• Multivariate Polynomial Values in Difference Sets