Well-distribution modulo one and the primes

TL;DR

The paper constructs an irrational number whose multiples with prime numbers are not uniformly distributed modulo 1, challenging assumptions about distribution properties of prime multiples.

Contribution

It proves the existence of an irrational number with non-uniform distribution of its prime multiples modulo 1, a novel result in distribution theory.

Findings

Existence of such an irrational number is established.

Prime multiples of this number are not well-distributed modulo 1.

Abstract

Let $(p_n)$ denote the sequence of prime numbers, with $2=p_1<p_2<\ldots$. We demonstrate the existence of an irrational number $\alpha$ having the property that the sequence $(\alpha p_n)$ is not well-distributed modulo $1$.