Variants of Romanoff's theorem

Artyom Radomskii

number theory

Variants of Romanoff's theorem

arXiv:2504.09954

summary

The paper investigates two sequences of positive integers and, under certain conditions, establishes a lower bound on how many integers up to a given size can be expressed as a sum of one element from each sequence.

Abstract

Let $A = {a_{n}}_{n = 1}^{\infty}$ and $B = {b_{n}}_{n = 1}^{\infty}$ be two sequences of positive integers (not necessarily distinct). Under some restrictions on $A$ and $B$ , we obtain a lower bound for a number of integers $n$ not exceeding $x$ that can be represented as a sum $n = a_{i} + b_{j}$ .

Topics & keywords

#additive number theory #romanoff's theorem #sumsets #integer sequences #lower boundsromanoff's theoremsumsetrepresentation functionlower boundadditive combinatorics