Balasubramanian 2001 additive complements squares

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theorem_1: Balasubramanian and Ramana's conditional lower bound for alpha, the liminf of b(N)/sqrt(N) over minimal additive complements of the squares up to N: if for a delta in (0,1) and all large N some minimal complement lies in [0, delta N], then alpha is at least an explicit function of delta that runs from 2 at delta = 0 to 4/pi at delta = 1.

theorem_p11: The generalization of Theorem 1 to p-th powers that Balasubramanian and Ramana record without proof in their concluding remarks: if for a delta in (0,1) and all large N some minimal additive complement of the p-th powers up to N lies in [0, delta N], then alpha(p) is at least an explicit function of p and delta.

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R. Balasubramanian, D. S. Ramana, Additive complements of the squares. C. R. Math. Rep. Acad. Sci. Canada 23 (2001), no. 1, 6-11.

Let b(N) be the least size of a set B in {0,...,N} such that every 1 <= n <= N is b + k^2 with b in B, and let alpha = liminf b(N)/sqrt(N); the trivial bound gives alpha >= 1 and the best unconditional bound the paper reports (p. 6), due independently to Habsieger and Cilleruelo, is alpha >= 4/pi. Theorem 1 proves a stronger explicit lower bound for alpha under the localization hypothesis that for some fixed delta in (0,1) and all large N some minimal complement lies inside [0, delta N]; the paper notes that the right-hand side is a continuous function of delta taking the value 2 at delta = 0 and 4/pi at delta = 1, so if for all large N some minimal complement has all elements o(N), then alpha >= 2, which with the inequality alpha <= 2 (see Zhai) gives alpha = 2 (p. 7). The proof counts representations with weights: Lemma 1 compares sum_B sum_k f(b+k^2) with sum_{1<=n<=N} f(n) for any f with f(t) >= 0 for t >= 0, and applying it to the monomials f_m(t) = t^m and integrating against the counting function of B gives, for each m >= 1, an inequality involving explicit kernels phi_m and g_m (the Corollary, p. 7); applied to minimal complements inside [0, delta N_k] along a sequence N_k with b(N_k)/sqrt(N_k) -> alpha, these give Theorem 1 as m tends to infinity, using the properties of g_m and phi_m in Propositions 1 and 2 (pp. 8--10). The paper presents Theorem 1 as an improvement over a result of Zhai obtained by a different method. Its concluding remarks (p. 11) record without proof the generalization to p-th powers, p >= 2: under the same localization hypothesis for minimal complements of the p-th powers, the liminf alpha(p) of b_p(N)/N^{1-1/p} is at least an explicit function of p and delta, which for p = 2 is the bound of Theorem 1.

Source: https://mathreports.ca/article/additive-complements-of-the-squares/. The copy read for this card, a scan of the printed pp. 6-11, prints "© Royal Society of Canada 2001." on its first page, every other right reserved.

Read status. Claims checked: Theorem 1 with the remark after it (p. 7) and the p-th power theorem (p. 11) were read clause by clause on the printed pages. The proof of Theorem 1 (pp. 7-10) was read but not checked step by step; the p-th power theorem has no proof in the paper.

Bears on. #33: every set A as in the problem yields additive complements of the squares up to N of size at most |A cap [0,N]| plus a constant, so any lower bound on alpha bounds the problem's liminf, and hence its limsup, from below; Theorem 1 gives such a bound only under its localization hypothesis, which the paper does not establish, so it adds nothing unconditional to 4/pi and does not determine the smallest limsup.

Results. Theorem 1 (p. 7); the p-th power theorem (p. 11, unnumbered). Lemma 1 and its Corollary (p. 7) are proof steps of Theorem 1, summarized on its page.

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