Konyagin 2009 erdos turan problem infinite groups

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corollary_1: Konyagin and Lev's corollary that every abelian group which is infinite with |2G| = |G|, or has prime exponent, has a basis of order two whose representation function is bounded by an absolute constant, except at zero when the exponent is 2.

theorem_1: Konyagin and Lev's classification for an infinite abelian group G with |2G| = |G|: G has a perfect basis of order two unless G is the direct sum of a group of exponent 3 and the group of order 2, and such a sum has no perfect basis but has a basis giving every element at most two representations up to the order of the summands.

theorem_2: Konyagin and Lev's theorem that every abelian group of exponent 2 has a basis of order two under which every non-zero element has at most 36 representations as a sum of two basis elements.

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Sergei V. Konyagin, Vsevolod F. Lev, The Erdős-Turán problem in infinite groups. arXiv preprint (2009). arXiv:0901.1649. Published in Additive Number Theory, Springer, New York, 2010, 195--202, doi:10.1007/978-0-387-68361-4_14. The copy read for this card is arXiv version v1 (12 January 2009). The arXiv record names arXiv's non-exclusive distribution license (arXiv:0901.1649), every other right reserved.

Konyagin and Lev completely settle the Erdős-Turán basis problem for infinite abelian groups G with |2G| = |G|. Theorem 1 shows such a G has a perfect basis (every element a sum of two basis elements in exactly one way, up to the order of the summands) unless G is an exponent-3 group plus a direct summand of order 2; such a G has no perfect basis, but has a basis in which every element has at most two representations, counted up to the order of the summands. Theorem 2 turns to groups of exponent 2, where for infinite G the element 0 has |G| representations as 2s under any basis: every abelian group of exponent 2 has a basis in which every non-zero element has at most 36 representations, proved by a construction adapted from the covering-radius-2 codes of Gabidulin, Davydov and Tombak [GDT91], via three hyperbola-like sets S_i = {(x, d_i/x)} in F x F. Corollary 1 combines these with Haddad-Helou and Ruzsa: every abelian group that is infinite with |2G| = |G|, or has prime exponent, has a basis whose representation function is at most one absolute constant, except at the zero element when the exponent is 2. The paper was read for problem 1192 mainly as an accessible summary of Ruzsa's paywalled 1990 work: the F_p x F_p basis with at most 18 representations when (2/p) = -1, and its corollary [R90, Theorem 1], from which every finite cyclic group easily gets a basis with representation function at most a constant not depending on the group's order, together with the Haddad-Helou and Nathanson group-side state of the art.

Source: https://arxiv.org/abs/0901.1649.

Bears on. #1192: the problem asks for bases of the natural numbers of each order r with mean-square representation count O(x); the paper's own results concern abelian groups and order two only and say nothing about bases of the natural numbers, and its link to the problem is its account (p. 2) of Ruzsa's 1990 work, the source of the problem's case r = 2, which it reports only for F_p x F_p and finite cyclic groups.

Results.

• Theorem 1 (p. 2): An infinite abelian G with |2G| = |G| has a perfect basis unless G is the direct sum of a group of exponent 3 and the group of order 2; such a G has no perfect basis but has a basis in which every element has at most two representations, counted up to the order of the summands.
• Theorem 2 (p. 3): Every abelian group of exponent 2 has a basis in which every non-zero element has at most 36 representations as a sum of two basis elements.
• Corollary 1 (p. 3): Every abelian group G that is either infinite with |2G| = |G| or of prime exponent has a basis whose representation function is bounded by an absolute constant independent of the group, except at the zero element when G has exponent 2.
• Lemma 1 (p. 3: an infinite abelian group of prime exponent p is isomorphic to F x F for an algebraically closed field F of characteristic p) and Lemma 2 (p. 4) are proof steps, summarized on the Theorem 1 and Theorem 2 pages.
• Cited (Ruzsa 1990, p. 2): For primes p with (2/p) = -1, F_p x F_p has a basis with at most 18 representations per element; Ruzsa's Theorem 1, derived from it, easily gives every finite cyclic group a basis with representation function at most a constant not depending on the group's order.

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