Cilleruelo 2013 dense sets integers prescribed representation functions

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corollary_1: Every f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least 1 is the order-2 representation function of a set A of integers with A(x) >> x^(sqrt 2 - 1 + o(1)).

corollary_2: For h >= 2 and epsilon > 0 there is g = g(h, epsilon) such that every f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least g is the order-h representation function of a set A of integers with A(x) >> x^(1/h - epsilon).

corollary_3: For every f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least 1 and every increasing omega tending to infinity, some set A has r_(A,h)(n) = f(n) for all integers n and lim sup of A(x) omega(x)/x^(1/h) positive.

theorem_1: For any f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least g, any B_h[g] sequence B and any decreasing epsilon(x) tending to 0, some set A of integers has r_(A,h)(n) = f(n) for every integer n and A(x) >> B(x epsilon(x)).

theorem_2: For any f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least g and any B_2[g] sequence B, some set A of integers has r_(A,2)(n) = f(n) for every integer n and A(x) >> B(x/3); the paper omits the proof.

theorem_3: There is a unique representation basis A of the integers with lim sup of A(x)/sqrt(x) at least 1/sqrt(2); the paper omits the proof.

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Javier Cilleruelo, Melvyn B. Nathanson, Dense sets of integers with prescribed representation functions. European Journal of Combinatorics 34 (2013), 1297–1306. arXiv:0708.2853, doi:10.1016/j.ejc.2013.05.012.

Theorem 1 states that for any function f from Z to N u {0, infinity} whose lower limit as |n| tends to infinity is at least g, any B_h[g] sequence B, and any decreasing epsilon(x) tending to 0, there is a set A of integers with r_(A,h)(n) = f(n) for all n and A(x) >> B(x epsilon(x)), where A(x) counts the elements with |a| <= x; so a prescribed representation function is achieved at the cost of only a slowly varying loss in the argument of B. The method spreads out a given B_h[g] sequence by inserting strings of zeros at fixed places in the binary expansions of its elements (the Inserting Zeros Transformation) and then adjoins a very sparse sequence to force exactly the prescribed multiplicities. Earlier, Nathanson had reached A(x) >> x^(1/(2h-1)) for f with lower limit at least 1, and Luczak and Schoen recovered that bound by enlarging B_h sequences with an extra Sidon-type property. Corollary 1 combines Theorem 1 with Ruzsa's Sidon set of density x^(sqrt 2 - 1 + o(1)) to realize any f with lower limit at least 1 by a set with A(x) >> x^(sqrt 2 - 1 + o(1)) for h = 2, which the paper says answers the third open problem of Chen's paper on unique representation bases, posed earlier by Nathanson; Corollary 2 uses Vu's B_h[g] sequences to get A(x) >> x^(1/h - epsilon) once g >= g(h, epsilon). Theorem 2 states an h = 2 form with density B(x/3), and Theorem 3 a unique representation basis with lim sup A(x)/sqrt(x) >= 1/sqrt(2), which the paper says answers Chen's first open problem; the proofs of both are omitted as very close to proofs in the authors' paper on perfect difference sets from Sidon sets. A remark after Theorem 3 uses Erdős's theorem on Sidon sets to show that liminf |A n (-x, x)|/sqrt(x) = 0 for every infinite Sidon set A of integers, which the paper reads as a negative answer to Chen's second open problem. Corollary 3 gives, for every f with lower limit at least 1 and every increasing omega tending to infinity, a set with order-h representation function f and positive lim sup of A(x) omega(x)/x^(1/h).

Source: https://arxiv.org/abs/0708.2853. The copy read for this card is arXiv:0708.2853v1, submitted 21 Aug 2007, not the journal article; the labels below are that preprint's. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:0708.2853), every other right reserved.

Read status. Claims checked: Theorems 1-3 and Corollaries 1-3 were read clause by clause on the printed pages. The proof of Theorem 1 (pp. 4-10) was read for structure only; the paper prints no proofs of Theorems 2 and 3.

Bears on. #158: the paper does not mention the problem. Theorems 1 and 2 with h = 2 and g = 2 take a B_2[2] sequence as input and return a set of integers, containing negative integers in general, with a prescribed representation function and density transferred from the input; they construct no B_2[2] subset of the natural numbers and do not address the lower limit of A(N)/N^(1/2) the problem asks about. The remark after Theorem 3 applies Erdős's theorem for Sidon sets, the one-representation case recorded as the problem's accepted partial claim.

Results. Labels and pages are those of arXiv:0708.2853v1.

• Theorem 1 (p. 2): prescribed order-h representation function with A(x) >> B(x epsilon(x)) for any B_h[g] sequence B.
• Corollary 1 (p. 2): order 2, lower limit of f at least 1, A(x) >> x^(sqrt 2 - 1 + o(1)).
• Corollary 2 (p. 3): order h, lower limit of f at least g(h, epsilon), A(x) >> x^(1/h - epsilon).
• Theorem 2 (p. 3): order 2, A(x) >> B(x/3) for any B_2[g] sequence B; proof omitted.
• Theorem 3 (p. 3): a unique representation basis with lim sup A(x)/sqrt(x) >= 1/sqrt(2); proof omitted.
• Corollary 3 (p. 4): order h, lower limit of f at least 1, and lim sup A(x) omega(x)/x^(1/h) > 0 for any increasing omega tending to infinity.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.