Diffraction of the primes and other sets of zero density

TL;DR

This paper demonstrates that the diffraction pattern of prime numbers is purely continuous with no sharp peaks, introduces a new concept of counting diffraction for zero-density sets, and explores various spectral types through examples.

Contribution

It introduces the notion of counting diffraction for zero-density sets and develops a comprehensive theory with multiple examples, extending classical diffraction concepts.

Findings

Diffraction of primes is absolutely continuous with no Bragg peaks.

Developed the theory of counting diffraction for zero-density sets.

Provided examples of sets with all possible spectral types.

Abstract

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.