Moreira's conjecture on sums and products in positive-density sets

Let ANA\subseteq\mathbb{N} have positive density, where its density is

d(A)=limNA{1,,N}Nd(A)=\lim_{N\to\infty}\frac{|A\cap\{1,\ldots,N\}|}{N}

whenever this limit exists. Moreira's conjecture. There exist x,yN\{1}x,y\in\mathbb{N}\backslash\{1\} such that

{x+y1,xy}A.\{x+y-1,xy\}\subseteq A.

This is a positive-density sum-product problem in the integers, motivated by analogous results over fields; the supplied text does not state whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Florian K. Richter, “Sums and products in sets of positive density”, arXiv:2507.00515 (2026).

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