Ding 2022 green s additive complement problem k

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theorem_1_1: Ding and Wang's theorem that for every integer k >= 2 and every additive complement B = {b_1, b_2, ...} of the k-th powers {1^k, 2^k, ...}, the limsup of (a_k n^{k/(k-1)} - b_n)/n is at least (k/(2(k-1))) Gamma(2 - 1/k)^2 / Gamma(2 - 2/k), where a_k = Gamma(2 - 1/k)^{k/(k-1)} Gamma(1 + 1/k)^{k/(k-1)}.

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Yuchen Ding, Li-Yuan Wang, Green's additive complement problem for k-th powers. Journal of the Korean Mathematical Society 59, no. 2 (2022), 299-309. doi:10.4134/JKMS.j210123.

Ding and Wang generalize Green's additive complement problem from squares to the k-th powers S^k = {1^k, 2^k, ...}. Theorem 1.1 (p. 301) states that for any integer k >= 2 and any additive complement B = {b_1, b_2, ...} of S^k, the limsup over n of (a_k n^{k/(k-1)} - b_n)/n is at least (k/(2(k-1))) Gamma(2 - 1/k)^2 / Gamma(2 - 2/k), where a_k = Gamma(2 - 1/k)^{k/(k-1)} Gamma(1 + 1/k)^{k/(k-1)} (Section 2 calls it a_k); the paper motivates a_k on p. 301 as the coefficient of the profile b_n ~ a_k n^{k/(k-1)} along which the average number of representations n = l^k + b tends to 1. For k = 2 the inequality is exactly the first author's earlier pi/4 bound (Remark 1.2), which the paper cites and does not prove again, and Example 1.3 evaluates the constant for k = 3 as approximately 0.684463. Conjecture 1.4 (p. 302) proposes that the limsup is +infinity for every k >= 2. For k > 2 the proof assumes the contrary, turns the resulting lower bound for b_n into an upper bound for the counting function B(n) by the binomial expansion, and with Euler-Maclaurin summation and Beta-function integrals shows that the total number of representations n = l^k + b with n <= N = K^k is at most N minus a positive multiple of N^{1-1/k}, plus O(N^{1-2/k}), which no additive complement allows for large K.

Source: https://doi.org/10.4134/JKMS.j210123. The copy read for this card is the journal's PDF, which prints "©2022 Korean Mathematical Society" at the foot of its first page (p. 299) and names no license; the journal's article page shows "© 2022. The Korean Mathematical Society." and an "Open Access" menu link but names no license for the article or the journal (https://jkms.kms.or.kr/journal/view.html?doi=10.4134/JKMS.j210123, read 2026-10-02), and the journal site, read the same day, states "A Copyright Transfer Agreement is required before the publication of a paper in this journal." and names no license, every other right reserved.

Bears on. #33: every additive complement of S^2 = {1, 4, 9, ...} is a set as in the problem, which allows n >= 0, though not conversely. For k = 2 Theorem 1.1 is the first author's earlier bound, cited and not proved again here: such a complement falls at least about (pi/4)n below the profile (pi^2/16)n^2, whose counting function is (4/pi)sqrt(N) + o(sqrt(N)), for infinitely many n. That deviation is 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; its cases k >= 3 concern higher powers, which the problem does not ask about.

Results. Theorem 1.1 (p. 301, with Remark 1.2 on pp. 301-302 and Example 1.3 and Conjecture 1.4 on p. 302).

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