An explicit economical additive basis

TL;DR

The paper constructs an explicit subset of natural numbers that covers all natural numbers as sums of two elements, yet has a very sparse structure in terms of the number of representations for large sums, answering a question posed by Erdős.

Contribution

It provides an explicit example of a subset of natural numbers that forms an additive basis with extremely sparse pairwise sums, addressing a longstanding open problem.

Findings

Constructed an explicit additive basis for natural numbers.

Showed the number of representations grows slower than any power of N.

Answered Erdős's question on sparse additive bases.

Abstract

We present an explicit subset $A\subseteq \mathbb{N} = \{0,1,\ldots\}$ such that $A + A = \mathbb{N}$ and for all $\varepsilon > 0$, \[\lim_{N\to \infty}\frac{\big|\big\{(n_1,n_2): n_1 + n_2 = N, (n_1,n_2)\in A^2\big\}\big|}{N^{\varepsilon}} = 0.\] This answers a question of Erd\H{o}s.