Effective bounds for Roth's theorem with shifted square common difference

TL;DR

This paper establishes new bounds on the size of subsets avoiding certain shifted square progressions, showing they are significantly smaller than the entire set, with bounds involving iterated logarithms.

Contribution

It provides the first effective bounds for Roth's theorem with shifted square common differences, answering a question posed by Green.

Findings

Sets avoiding the specified progressions are of size at most N divided by a logarithm iterated m times.

The bounds are explicit and involve iterated logarithmic functions.

This advances understanding of additive combinatorics related to polynomial progressions.

Abstract

Let $S$ be a subset of $\{1,\ldots,N\}$ avoiding the nontrivial progressions $x, x+y^2-1, x+ 2(y^2-1)$. We prove that $|S|\ll N/\log_m{N}$, where $\log_m $ is the $m$-fold iterated logarithm and $m\in\mathbf{N}$ is an absolute constant. This answers a question of Green.