A Romanoff-type theorem for a multiset of products of powers

Abstract

Let b₁,…,b_d≥2 be fixed integers, and let k be their multiplicative rank. We study representations n=a+b₁^u₁⋯ b_d^u_d, where a belongs to a set 𝒜 of positive integers and u₁,…,u_d are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on 𝒜, we obtain a lower bound for the number of integers with many such representations, in terms of d and k. In particular, when 𝒜 is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers n≤ x have at least c₁(log x)^d−1 or c₁(log x)^d−1/2 representations, respectively, for some c₁>0 and all sufficiently large x. No multiplicative independence or coprimality assumptions on the bases are required.

A necessary condition for a Romanoff type theorem
Xiao-Hui Yan, Ya-Ting Xu
arXiv20 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
The Beurling-type theorem for powers of weighted Bergman shifts
Shibo Xu, Tao Yu
arXiv28 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
On a conjecture on Romanoff type representations
Xiao-Hui Yan
arXiv9 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Romanoff type theorems for polynomials over finite fields
Enci Wang
arXiv19 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
S_n-equivariant L-classes of Cartesian powers of Witt spaces and the Hodge index conjecture for symmetric products
Mohammadali Aligholi
arXiv2 Nov 2026PreprintCC BY 4.0Full textAlgebra & number theory Geometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{radomskii2026romanofftype,
  title = {{A Romanoff-type theorem for a multiset of products of powers}},
  author = {Artyom Radomskii},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38207},
  eprint = {2609.38207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38207},
}

RIS

TY  - UNPB
TI  - A Romanoff-type theorem for a multiset of products of powers
AU  - Artyom Radomskii
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.38207
UR  - https://arxiv.org/abs/2609.38207
AB  - Let b_1,…,b_d≥2 be fixed integers, and let k be their multiplicative rank. We study representations n=a+b_1^u_1⋯ b_d^u_d, where a belongs to a set 𝒜 of positive integers and u_1,…,u_d are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on 𝒜, we obtain a lower bound for the number of integers with many such representations, in terms of d and k. In particular, when 𝒜 is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers n≤ x have at least c_1(log x)^d−1 or c_1(log x)^d−1/2 representations, respectively, for some c_1>0 and all sufficiently large x. No multiplicative independence or coprimality assumptions on the bases are required.
ER  -

Abstract

Related

Cite