On van der Corput property of shifted primes

Abstract

Abstract. We prove that the upper bound for the van der Corput property of the set of shifted primes is O((log n) −1+o(1) ), giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes p − 1. We construct normed non-negative valued cosine polynomials with the spectrum in the set p − 1, p ≤ n, and a small free coefficient a 0 = O((log n) −1+o (1) ). This implies the same bound for the intersective property of the set p − 1, and also bounds for several properties related to uniform distribution of related sets.

Ergodic characterization of van der Corput sets

“…In particular, for the set of squares, γ(n) = O((log n) −1/3 ) [10], and for the set of shifted primes p − 1, γ(n) = (log n) −1+o (1) [11].…”