A Sharp Bound for Additive Complements

Zihan Zhang

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arXiv · 2610.29963

A Sharp Bound for Additive Complements

Zihan Zhang

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Abstract

Let $A$ and $B$ be infinite sets of nonnegative integers whose sum contains all sufficiently large integers, and let $A(x)$ and $B(x)$ be their counting functions. We give a discrete proof of the sharp implication \[ \limsup_{x\to\infty}\frac{A(x)B(x)}{x}<\frac32 \quad\Longrightarrow\quad A(x)B(x)-x\longrightarrow+\infty. \] The constant $3/2$ is best possible. Using finite convolution sums, we show that a long gap in one set forces a gap in the other. We then compare the counting functions before and after the gap to obtain the sharp bound.

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BibTeX RIS

Zihan Zhang. 2026-10-21. A Sharp Bound for Additive Complements. https://arxiv.org/abs/2610.29963

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