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Source. Theorem 1.7, p. 5, with Theorems 3.1 (p. 7) and 4.2 (p. 10), of Javier Cilleruelo, Imre Z. Ruzsa and Carlos Vinuesa, Generalized Sidon sets, Advances in Mathematics 225 (2010), 2786--2807, arXiv:0909.5024. Labels and pages are those of arXiv:0909.5024v1 (28 Sep 2009), the edition named on the source card.

Read depth. Claims checked: the three statements and the definitions they use were read clause by clause on the page images; the proofs (Sections 3--4, pp. 7--11) were read for structure only. Nothing here is independently reviewed.

Statement

Setting (pp. 1--4). A set $A$ in a commutative group is a $g$-Sidon set if every $x$ has at most $g$ representations $x=a_1+a_2$ as an ordered pair $(a_1,a_2)\in A^2$ (Definitions 1.1 and 1.2, pp. 1--2). For a finite commutative group $G$, $\alpha_g(G)$ is the largest size of a $g$-Sidon set $A\subset G$, $\alpha_g(q)=\alpha_g(\mathbb Z_q)$ (Definition 1.6, p. 4), and

$$\alpha_g=\limsup_{q\to\infty}\frac{\alpha_g(q)}{\sqrt q}\qquad\text{(p. 4)}.$$

The paper notes the obvious bound $\alpha_g(q)\le\sqrt{gq}$ (p. 4).

Theorem 1.7 (p. 5). We have $\alpha_g=\sqrt g+O\!\left(g^{3/10}\right)$; in particular $\lim_{g\to\infty}\alpha_g/\sqrt g=1$.

Theorem 3.1 (p. 7). For each $k$ and every sufficiently large prime $p\ge p_0(k)$, there is a set $A\subseteq\mathbb Z_p^2$ with $kp-k+1$ elements that is a $g$-Sidon set for $g=\lfloor k^2+2k^{3/2}\rfloor$.

Theorem 4.1 (p. 10). If $A\subseteq\mathbb Z_p^2$ is a $g$-Sidon set with $\lvert A\rvert=m$ and $q=p^2s$ with $s$ a positive integer, there is a $g'$-Sidon set $A'\subseteq\mathbb Z_q$ with $\lvert A'\rvert=ms$ and $g'=g(s+1)$.

Theorem 4.2 (p. 10). For any positive integers $k,s$ and every sufficiently large prime $p$, there is a set $A\subseteq\mathbb Z_{p^2s}$ with $(kp-k+1)s$ elements that is a $\lfloor k^2+2k^{3/2}\rfloor(s+1)$-Sidon set.

Proof pointer

Theorem 3.1 (pp. 7--9) takes the union of the $k$ parabolas $\{(x,x^2/u):x\in\mathbb Z_p\}\subset\mathbb Z_p^2$ for $u=t+1,\ldots,t+k$. Lemma 3.2 (p. 7), a Legendre-symbol identity for the representation counts of two parabolas, bounds $r(x)$ by $k^2$ plus a character sum in $t$, and an average over $t$ finds a $t$ for which that sum is small. Theorem 4.1 (p. 10) maps $(a,b)$ to the integers $a+cp+bsp$, $0\le c\le s-1$, modulo $p^2s$; Theorem 4.2 combines the two. On p. 11 the paper takes $g=\lfloor k^2+2k^{3/2}\rfloor(s+1)$ with $k=4s^2$, so $s=\Theta(g^{1/5})$, and the prime number theorem gives $\alpha_g/\sqrt g\ge1+O(g^{-1/5})$ for these $g$. The matching upper bound $\alpha_g\le\sqrt g$ is the obvious estimate of p. 4, which p. 11 does not restate, and the page does not write out the passage from these values of $g$ to all $g$.

Dependencies

None from the corpus; Weil's bound for character sums (p. 9) and the prime number theorem (p. 11).

Bears on

The theorem concerns finite cyclic groups and bears on no Erdős problem directly. Its construction (Theorem 4.2) feeds the lower bound of Theorem 1.5.