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Statement

Setting (p. 1). A basis of order two of an abelian group is a subset $S$ such that every element is a sum of two elements of $S$; representations are counted as ordered pairs.

Theorem 2 (p. 3, quoted). "Each abelian group of exponent 2 possesses a basis such that every non-zero element of the group has at most 36 representations as a sum of two elements of this basis."

The theorem covers finite and infinite groups of exponent 2 with one constant. Zero is excluded because in exponent 2 it equals $s+s$ for every $s$ in the basis: the paper notes (pp. 2-3) that no infinite family of exponent-2 groups, even of finite groups, has bases with uniformly bounded representation functions, and (p. 3) that excluding zero in exponent 2 is the same as disregarding representations with equal summands.

Source. Sergei V. Konyagin and Vsevolod F. Lev, The Erdős-Turán problem in infinite groups, arXiv:0901.1649v1 (2009); published in Additive Number Theory, Springer, New York, 2010, 195--202. Labels and pages here are those of arXiv v1: Theorem 2 and Lemma 1 on p. 3, the proof on pp. 6-8. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read but not checked step by step; the passage from the bound 18 for $\mathbb F\times\mathbb F$ below to the stated 36 for every group of exponent 2 is not written out on pp. 6-8 and was not reconstructed here. Nothing here is independently reviewed.

Proof pointer

Pages 6-8. The construction is adapted from the covering-radius-2 codes of Gabidulin, Davydov and Tombak (IEEE Trans. Inform. Theory 37 (1991), 219-224). Using Lemma 1 (p. 3), the paper reduces to showing that for a field $\mathbb F$ of characteristic 2, finite or algebraically closed, with $|\mathbb F|>2$, the group $\mathbb F\times\mathbb F$ has a basis with representation function at most 18 away from $(0,0)$. It fixes non-zero $d_1,d_2,d_3$ with $d_1+d_2+d_3=0$ and takes $S=S_1\cup S_2\cup S_3$ with $S_i=\{(x,d_i/x):x\in\mathbb F^\times\}$. Each of the nine counts $r_{ij}(u,v)$ of representations from $S_i+S_j$ counts roots of a quadratic that is not identically zero unless $(u,v)=(0,0)$, so is at most 2. That every non-zero $(u,v)$ is represented is checked in three cases, the last, for finite $\mathbb F$, using that $x\mapsto x+x^2$ has image of codimension 1 over $\mathbb F_2$ together with $d_1+d_2+d_3=0$.

Dependencies

Lemma 1 (p. 3) of the same paper; the construction of Gabidulin, Davydov and Tombak cited above.

Bears on

• Problem 1192: the problem concerns bases of $\mathbb N$ of order $r$ with $\sum_{n\le x}f_r(n)^2\ll x$. Theorem 2 concerns groups of exponent 2 and order two only; it says nothing about bases of $\mathbb N$.