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Statement

Setting (p. 146). $p$ is an odd prime, $\mathbb Z_p=\mathbb Z/p\mathbb Z$, and $G=\mathbb Z_p^2$. For $k\in\mathbb Z_p$, $Q_k=\{(u,ku^2):u\in\mathbb Z_p\}\subset G$. For $B\subset G$, $\sigma_B(g)$ and $\delta_B(g)$ count the ordered pairs $(x,y)\in B^2$ with $x+y=g$ and $x-y=g$ respectively, as $\sigma$ and $\delta$ do for integers (p. 145).

Lemma 2.2 (p. 147, quoted). "Assume $\left(\frac{2}{p}\right)=-1$ and put $B=Q_3\cup Q_4\cup Q_6$. We have $B+B=G$, $\sigma_B(g)\leqslant18$ for all $g\in G$ and $\delta_B(g)\leqslant18$ for all $g\neq0$."

Each $Q_k$ has $p$ elements, so $|B|\le3p$. Konyagin and Lev (p. 2) cite this result as Ruzsa's basis of $\mathbb F_p\times\mathbb F_p$ in which every element has at most 18 representations as a sum of two basis elements.

Source. Imre Z. Ruzsa, A Just Basis, Monatsh. Math. 109 (1990), 145--151, doi:10.1007/BF01302934. Labels and pages are those of the journal print: the setting and Lemma 2.1 on p. 146, Lemma 2.2 on p. 147, its proof on pp. 147--148. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 147--148. Lemma 2.1 (p. 146) counts the solutions of $g=x+y$ with $x\in Q_k$, $y\in Q_l$, $k,l\ne0$: if $k+l\ne0$ there are at most two, and for $g=(a,b)$ a solution exists unless $\left(\frac{(k+l)b-kla^2}{p}\right)=-1$; if $k+l=0$ there is at most one unless $g=0$, which has $p$. An element missing from both $Q_4+Q_4$ and $Q_3+Q_6$ would make two such symbols equal to $-1$, and their product equals $\left(\frac{2}{p}\right)=-1$, a contradiction; so $B+B=G$. For the counts, placing $x$ and $y$ in $Q_3$, $Q_4$ or $Q_6$ gives nine sub-equations with at most two solutions each.

Dependencies

Lemma 2.1 (p. 146).

Bears on

No Erdős problem directly. It is the modular input to Theorem 1. Konyagin and Lev describe it in their introduction (p. 2); their Corollary 1 draws on Theorem 1, which they call its corollary, and on Haddad and Helou's extension to $\mathbb F\times\mathbb F$, not on this lemma directly.