Variants of Romanoff's theorem

TL;DR

This work generalizes Romanoff's theorem by bounding the number of integers $n\le x$ representable as $n=a_i+b_j$ for two positive-integer sequences $\mathcal{A}$ and $\mathcal{B}$ under natural growth and regularity assumptions. A central second-moment argument yields a universal lower bound, expressed as $\#\{n\le x: r(n)\ge c_1\mathcal{B}(x)/\eta(x)\} \gg x\mathcal{B}(x)/(\mathcal{B}(x)+\rho_{\mathcal{B}}(x)\eta(x))$, with $\mathcal{B}(x)$ capturing the size of the $\mathcal{B}$-sequence and $\rho_{\mathcal{B}}(x)$ its maximal multiplicity. The paper then demonstrates powerful corollaries for various natural $\mathcal{A},\mathcal{B}$, including primes, sums of two squares, and polynomial-exponential families $\mathcal{B}=\{a^{f(n)}\}$, as well as for sums involving elliptic curves (non-CM) and Euler- totients, using Brun–Titchmarsh-type estimates and residue-class considerations. Altogether, it provides a unified framework for establishing positive-density lower bounds on additive representations across classical sets and arithmetic-geometric objects. The results significantly broaden the scope of Romanoff-type density statements and connect additive representation problems to elliptic-curve arithmetic and totient-weighted sums.

Abstract

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

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Corollary 1.2
• Theorem 1.4
• Corollary 1.3
• Theorem 1.5
• Corollary 1.4
• Theorem 1.6