Random linear configurations in dense sets and primes

2026/07/30 by Lasse Grimmelt, Joni Teräväinen
Mathematics · #math.NT #math.CO

paper · pdf

58 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We prove that every polylogarithmically dense subset of [N] contains a nontrivial configuration x+b₁m,…,x+b_km for almost all choices of the coefficient vector (b₁,…, b_k) in a wide range of scales. We prove the same statement for polylogarithmically relatively dense subsets of the primes, in a shorter range of scales. The main ingredients are a new quantitative generalised von Neumann theorem, degree lowering to the U¹⁺ norm, and densification arguments that transfer the result to the primes.

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