A discretised projection theorem in the plane

Abstract

The main result of this paper is that for any $1/2\le s<2-\sqrt{2}\approx 0.5858$, there is a number $\sigma =\sigma (s)<s$ with the following property. Let $\delta >0$ be small, assume that $A\subset [0,1]$ is a $(\delta ,1/2)$-set, and that $E\subset [0,1]$ contains $\gtrsim {\delta }^{-\sigma }$ roughly ${\delta }^s$-separated points. Then there exists a number $t\in E$ such that $A+tA$ contains $\gtrsim {\delta }^{-s}$ $\delta$-separated points. For $\sigma =s$, this is essentially a consequence of Kaufman's well-known bound for exceptional sets of projections. Our proof consists of a structural observation concerning sets, for which Kaufman's bound is near-optimal, combined with (an adaptation of) Solymosi's argument for his "$4/3$" sum-product theorem.

15 pages, 3 figures. v2: slight improvement to main theorem

Cited by in corpus (1)

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