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Statement

Setting. On p. 1 a sequence of positive integers is a Sidon sequence when all sums $a+a'$ with $a\le a'$ in it are distinct. The paper uses the modular terms without a separate definition: a Sidon set in $\mathbb{Z}_N$ is one whose pairwise sums, taken in the group, are distinct apart from the order of the summands, and $S$ is a basis of order $3$ in $\mathbb{Z}_N$ when every element of $\mathbb{Z}_N$ is a sum of three elements of $S$.

Theorem 1.1 (p. 1, quoted). "For all $N$ large enough, the cyclic group $\mathbb{Z}_N$ contains a Sidon set $S\subset\mathbb{Z}_N$ which is a basis of order $3$ in $\mathbb{Z}_N$."

The paper presents it as the modular version of its Conjecture 1.1 (p. 1), Erdős's conjecture that there is a Sidon basis of order $3$ of the positive integers.

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 and its definitions were read clause by clause on the page images. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Sections 2.1 and 2.2 (pp. 6--10). For $N$ large, take a prime $p\equiv1\pmod3$ with $4p^2<N<5p^2$ and the Erdős-Turán Sidon set $\{x+(x^2)_p(2p):0\le x\le p-1\}$, where $(y)_p$ is the least non-negative residue of $y$ modulo $p$; it lies in $[0,N/2)$, so it is also Sidon in $\mathbb{Z}_N$ (p. 8). The basis property reduces to representing every integer $r_1+r_2(2p)$ in a window of length $5p^2$ (equation (2.11), p. 9). Proposition 2.1 (p. 6), deduced from a weak form of a theorem of Granville, Shparlinski and Zaharescu stated as Theorem 2.2 (p. 6), says that the points $((x_1)_p/p,(x_2)_p/p,(x_1^2)_p/p,(x_2^2)_p/p)$ with $(x_1,x_2)$ on the conic $x_1^2+x_2^2+(x_1+x_2-r_1)^2\equiv r_2\pmod p$ are well distributed in $[0,1]^4$ as $p\to\infty$; a point in a suitable box then gives the three summands without carries (pp. 9--10).

The weaker Theorem 2.1 (p. 4), that infinitely many $\mathbb{Z}_N$ contain a Sidon set over which every element is a sum of three pairwise distinct elements, has a separate proof through Ruzsa's set $\{(x,g^x)\}$ in $\mathbb{Z}_{p-1}\times\mathbb{Z}_p$ and Hasse's bound for an elliptic curve (pp. 4--5). Sections 4.1 and 5.1 (pp. 14, 18) fix the set $S$ of the proofs of Theorem 1.2 and Theorem 1.3 from it, while p. 5 names Corollary 2.1 as the input to the proof of Theorem 1.3.

Bears on

• Problem 157: the problem asks for an infinite Sidon set of integers that is an asymptotic basis of order 3, the paper's Conjecture 1.1. This theorem proves the analogue in the finite cyclic groups $\mathbb{Z}_N$ for all large $N$; it says nothing about sets of integers and does not settle the problem.