number theory

Random linear configurations in dense sets and primes

arXiv:2607.28091

summary

The paper shows that any subset of the integers up to N that is dense at a polylogarithmic level contains nontrivial linear configurations of the form x+b₁m,…,x+b_km for most coefficient choices, and extends a similar result to relatively dense subsets of the primes.

Abstract

We prove that every polylogarithmically dense subset of contains a nontrivial configuration for almost all choices of the coefficient vector 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 norm, and densification arguments that transfer the result to the primes.

58 pages

Topics & keywords

#additive combinatorics#linear configurations#dense subsets#primes#uniformity normspolylogarithmic densitygeneralized von Neumann theoremU^{1+} normdensificationlinear patterns
Random linear configurations in dense sets and primes · wovepaper