Ding 2022 note additive complements squares

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Yuchen Ding, Yu-Chen Sun, Li-Yuan Wang, Yutong Xia, A note on additive complements of squares. arXiv:2211.16810 (2022). Published as "A note on additive complements of the squares", Discrete Mathematics 349 (2026), no. 2, 114763, doi:10.1016/j.disc.2025.114763. The copy read for this card is arXiv:2211.16810v3.

For the squares S and any additive complement W, Theorem 1.1 shows that for large N the sum of R_{S,W}(n) over n <= N exceeds N by at least 0.193 N^{1/2}, improving the lower bound of order N^{1/4} log N that follows from Chen and Fang's Theorem 1.1 to one of order N^{1/2}. Theorem 1.2 feeds this into Green's formulation and improves Ding's earlier deviation bound pi/4 to limsup_n ((pi^2/16)n^2 - w_n)/n >= pi/4 + 0.193 pi^2/8. The proof refines Chen and Fang's argument by using only solutions of x^2 + d_1 = y^2 + d_2 < N with |x - y| large: this yields >> 1 solutions instead of >> log N, but lets d_1, d_2 range over W cap [epsilon_0 N] instead of W cap [c N^{1/2}]. For Erdős problem 33 this is a citation-trail paper: it quantifies how far any complement of the squares must be from exact-on-average and strengthens the linear-scale deviation in Green's problem, constraining near-exact complements without determining problem 33's optimal limsup constant.

Source: https://arxiv.org/abs/2211.16810. The arXiv record (https://arxiv.org/abs/2211.16810, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #33

Results to transcribe.

• Theorem 1.1: For any additive complement W of the squares and sufficiently large N, sum_{n<=N} R_{S,W}(n) - N >= 0.193 N^{1/2}, improving the bound

  N^{1/4} log N that follows from Chen-Fang's Theorem 1.1.

• Theorem 1.2: For any additive complement W = {w_n} of the squares, limsup_n ((pi^2/16)n^2 - w_n)/n >= pi/4 + 0.193 pi^2/8.
• Lemma 2.1: Let delta, delta_0 > 0 satisfy delta^2 + delta_0 <= 1 and delta_0^2/(16 delta^2) + delta_0 < 1, let K = floor(delta N^{1/2}) be a positive integer, and let D be a set of nonnegative integers with 4K | d - d' for all d, d' in D. Then for all sufficiently large N, the sum of R_{S,D}(n) - 1 over n <= N with R_{S,D}(n) >= 1 is at least D(delta_0 N) - 2.