On a Romanoff type problem of Erd\H{o}s and Kalm\'{a}r

Yuchen Ding

On a Romanoff type problem of Erd\H{o}s and Kalm\'{a}r

Number Theory 2025-09-09 v2

Authors: Yuchen Ding

Abstract

Let $\mathbb{N}$ and $\mathcal{P}$ be the sets of natural numbers and primes, respectively. Motived by an old problem of Erd\H os and Kalm\'ar, we prove that for almost all $y>1$ the lower asymptotic density of integers of the form $p+\lfloor y^k\rfloor~(p\in \mathcal{P},k\in \mathbb{N})$ is positive.

Keywords

asymptotic density prime numbers floor function Diophantine approximation Erdős–Kalmár problem

Cite

@article{arxiv.2503.22700,
  title  = {On a Romanoff type problem of Erd\H{o}s and Kalm\'{a}r},
  author = {Yuchen Ding},
  journal= {arXiv preprint arXiv:2503.22700},
  year   = {2025}
}

Comments

the expositions of the introduction are substantially extended

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