Pointwise ergodic theorem along primes of the form x² + ny²

2025/08/21 by Fornal, Jan · 4 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.15466

Abstract

This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form x² + ny². Following the influential paper of Bourgain \citebourgain1989pointwise, we employ the Hardy-Littlewood circle method where major arc and minor arc estimates for the set of prime ideals constitute the main novelty of the paper. We also prove that our convergence results cannot be extended to class of L¹ functions.

Citations

• On a multi-parameter variant of the Bellow-Furstenberg problem
• Discorrelation between primes in short intervals and polynomial phases
• Divergent Square Averages
• Discrete analogues in harmonic analysis: Spherical averages

Cited by

• Ramanujan's function on small primes
• Pointwise convergence of ergodic averages along quadratic bracket polynomials
• Isogeny Graphs in Superposition and Quantum Onion Routing
• Equilateral triangular waveguides, modal structure, attenuation characteristics and quality factor

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