Chen 2017 additive complements squares

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corollary_1_1: Chen and Fang's corollary that every additive complement B = {b_n} of the squares S = {1, 4, 9, ...} has limsup of ((pi^2/16)n^2 - b_n)/(n^(1/2) log n) at least sqrt(2/pi)/log 4, and their conjecture that this limsup is infinite.

corollary_1_2: Chen and Fang's corollary that the set of floor((pi^2/16)n^2), n = 1, 2, ..., is not an additive complement of the squares S = {1, 4, 9, ...}.

theorem_1_1: Chen and Fang's theorem that for every additive complement B of the squares S = {1, 4, 9, ...}, the sum of R_{S,B}(n) over n up to N exceeds N by at least c B(2 sqrt N) log B(2 sqrt N) for all large N, with c a positive constant, so the excess tends to infinity.

theorem_1_2: Chen and Fang's theorem that for positive constants alpha < sqrt(2/pi)/log 4 = 0.5755... and beta, a sequence B = {b_n} with b_n at least (pi^2/16)n^2 - alpha n^(1/2) log n - beta n^(1/2) for every n >= 1 is not an additive complement of the squares S = {1, 4, 9, ...}.

theorem_2_1: Chen and Fang's lower bound, for any infinite sequence D of nonnegative integers, on the excess sum over n at most X with R_{S,D}(n) >= 1 of R_{S,D}(n) - 1, where S is the squares from 1: it is at least (1+o(1))/log 4 times D(2 sqrt X) log D(2 sqrt X) for all large X.

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Yong-Gao Chen, Jin-Hui Fang, Additive complements of the squares. Journal of Number Theory 180 (2017), 410–422. doi:10.1016/j.jnt.2017.04.016.

Prompted by a question Ben Green put to the second author, whether some additive complement B of the squares S has b_n = (pi^2/16)n^2 + o(n^2), Theorem 1.1 proves that for any additive complement B of S, the excess sum_{n<=N} R_{S,B}(n) - N is at least cB(2sqrt(N))log B(2sqrt(N)) for large N and in particular tends to infinity; it is deduced from the more general Theorem 2.1 valid for arbitrary infinite sequences D of nonnegative integers. Theorem 1.2 turns this into a lower-bound obstruction on the elements themselves: if b_n >= (pi^2/16)n^2 - alpha*n^{1/2}log n - betan^{1/2} for all n >= 1, with positive constants alpha < sqrt(2/pi)/log 4 = 0.5755... and beta, then B is not an additive complement of S. Corollary 1.1 states the resulting lower bound sqrt(2/pi)/log 4 for the lim sup of ((pi^2/16)n^2 - b_n)/(n^{1/2} log n), and Corollary 1.2 concludes that the explicit set {floor((pi^2/16)n^2)} is not an additive complement of the squares. None of these results decides Green's question, whose o(n^2) error term allows larger deviations. The paper situates this against the classical Erdős-Moser problem on alpha = lim inf alpha(N)/sqrt(N), where alpha(N) is the least size of a set B_N of nonnegative integers with every positive n <= N of the form b + k^2 (b in B_N); it records alpha >= 4/pi as the best known bound and credits it independently to Cilleruelo, Habsieger, and Balasubramanian-Ramana. The paper closes its introduction with the conjecture that the lim sup of Corollary 1.1 is +infinity (p. 413).

Source: https://doi.org/10.1016/j.jnt.2017.04.016. The copy read for this card is the publisher's typeset article, which prints "© 2017 Elsevier Inc. All rights reserved." on its first page, every other right reserved.

Read status. Claims checked: Theorems 1.1, 1.2 and 2.1 and Corollaries 1.1 and 1.2 were read clause by clause on the printed pages. The proofs (pp. 413-422) were read but not checked step by step.

Bears on. #33: every additive complement of S = {1, 4, 9, ...} is a set as in the problem, which allows n >= 0, though not conversely; for such complements Theorem 1.2 and its corollaries show that the terms cannot satisfy b_n >= (pi^2/16)n^2 - alpha n^{1/2} log n - beta n^{1/2} for all n >= 1 when beta > 0 and 0 < alpha < sqrt(2/pi)/log 4, where (pi^2/16)n^2 is the profile whose counting function is (4/pi)sqrt(N) + o(sqrt(N)). These deviations are of lower order, so the paper does not raise the lower bound 4/pi for either quantity the problem asks about and does not determine the smallest lim sup.

Results. Theorem 1.1 (p. 412); Theorem 1.2 (p. 413); Corollary 1.1 (p. 413, with the paper's conjecture); Corollary 1.2 (p. 413); Theorem 2.1 (p. 414). Lemma 2.1 (p. 413) is a proof step of Theorem 2.1, summarized on its page.

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