On van der Corput property of shifted primes

TL;DR

This paper establishes an upper bound for the van der Corput property of shifted primes, showing it diminishes at a rate of approximately 1 over log n, and constructs specific cosine polynomials to demonstrate this.

Contribution

It provides the first explicit upper bound for the van der Corput property of shifted primes and constructs associated cosine polynomials with controlled spectral properties.

Findings

Upper bound for van der Corput property: O((log n)^{-1+o(1)})

Constructed cosine polynomials with spectrum in shifted primes

Bound for Poincaré property and related uniform distribution properties

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 Poincar\'e property of the set p-1, and also bounds for several properties related to uniform distribution of related sets.