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Statement

Setting (p. 2, Definition 2). For $0<\varepsilon<1$, a set $A$ is an asymptotic basis of order $h+\varepsilon$ when every sufficiently large positive integer $n$ is a sum of $h+1$ elements of $A$, one of them at most $n^\varepsilon$:

$$n=a_1+\cdots+a_{h+1},\quad a_1,\dots,a_{h+1}\in A,\quad a_{h+1}\leqslant n^\varepsilon.$$

The paper's words are "one of them smaller than $n^\varepsilon$", and the display has $\leqslant$. A Sidon basis of order $h+\varepsilon$ is such a basis that is also a Sidon sequence, all sums $a+a'$ with $a\le a'$ in $A$ being distinct (p. 1).

Theorem 1.3 (p. 2, quoted). "For any $\varepsilon>0$ there exists a Sidon basis of order $3+\varepsilon$." The paper restates it on the same page: for every $\varepsilon>0$ some Sidon sequence $A$ of positive integers has every sufficiently large positive integer $n$ of the form

$$n=a_1+a_2+a_3+a_4,\quad a_1,a_2,a_3,a_4\in A,\quad a_4\leqslant n^\varepsilon.$$

Definition 2 restricts $\varepsilon$ to $(0,1)$ while the theorem says $\varepsilon>0$; for $\varepsilon\ge1$ the condition $a_4\le n^\varepsilon$ holds for every representation of $n$. The paper calls this its strongest approximation to Conjecture 1.1 (p. 2) and recalls earlier Sidon bases of order 7 (Deshouillers and Plagne) and order 5 (Kiss); its note of 23 April 2013 (p. 3) reports that Kiss, Rozgonyi and Sándor independently obtained a Sidon sequence that is an asymptotic basis of order 4.

Source. J. Cilleruelo, On Sidon sets and asymptotic bases, Proceedings of the London Mathematical Society 111 (2015), 1206--1230, read in arXiv:1304.5351v2 (titled "Sidon basis") as identified on the source card; labels and pages are that preprint's.

Read depth. Claims checked: the statement, Definition 2 and the strategy of Section 5.1 were read clause by clause on the page images. The proof and the expected-value computations of Section 6.2 were not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 5 (pp. 18--21), with expected values in Section 6.2 (pp. 25--31). The proof follows that of Theorem 1.2 in the space $\mathcal{S}_m(\gamma;S\bmod N)$ of Definition 3 (p. 11) with $\gamma=2/3+\varepsilon/(9+9\varepsilon)$; any $\gamma$ with $(2+3\varepsilon)/(3+4\varepsilon)<\gamma<(2+\varepsilon)/(3+\varepsilon)$ would do (p. 18). Section 5.1 takes $S$ from Theorem 2.1, while p. 5 says that Corollary 2.1 (p. 5), the four-summand version with pairwise distinct summands, is the input to this proof. The Sidon lifting process of Definition 7 (p. 18) deletes every element involved in a repeated sum. Proposition 5.1 (p. 19) gives, with probability 1, at least a constant times $n^{2\varepsilon^2/(9+9\varepsilon)}$ representations of each large $n$ as a sum of four elements in distinct classes with the least at most $n^\varepsilon$, and Proposition 5.2 (p. 19) bounds the destroyed ones by a constant with probability $1-o_m(1)$.

Bears on

• Problem 157: the problem asks for an infinite Sidon set that is an asymptotic basis of order 3. This theorem gives a Sidon sequence in which every large $n$ is a sum of four elements, one of them at most $n^\varepsilon$, a weaker property than being an asymptotic basis of order 3; it does not settle the problem.