Ding 2025 cross representations additive complements r th

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Yuchen Ding, Ben Krause, Csaba Sándor, Yu-Chen Sun, Zihan Zhang, Cross representations of additive complements of r-th powers. arXiv preprint (2025). arXiv:2512.15407.

Motivated by a 1993 conjecture of Cilleruelo, the authors bound below the excess of the cross-representation count f_r(n) = #{(w, m^r) : n = w + m^r} for an additive complement W_r of the r-th powers. Theorem 1 shows that if S has counting function S(x) ~ c_r x^{1/r} then for any additive complement W the sum over n <= N of f(n) minus N is at least of order N^{1-1/r}; previously this was known only for r = 2. Theorem 2 sharpens the square case to a bound of order N^{3/4}/sqrt(T(N)), i.e. N^{3/4-o(1)}, improving the N^{1/2} bound of Ding, Sun, Wang and Xia. The methods are Abel summation together with bipartite-graph counting and input from the multiplication table problem. For Erdős problem 33 this is the current state of the art on how far a complement of the squares must be from exact-on-average; the earlier citation metadata under the title 'No exact on average additive complements of squares' refers to a substantially different v1 whose stronger linear-excess claim is absent here, and the present version does not settle the density constant in problem 33.

Source: https://arxiv.org/abs/2512.15407. The copy read for this card is arXiv:2512.15407v6 (stamped 9 July 2026); Theorems 1 and 2 are on p. 3. The arXiv record (https://arxiv.org/abs/2512.15407, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #33

Results to transcribe.

• Theorem 1: For S with S(x) ~ c_r x^{1/r} and any additive complement W of S, sum_{n<=N} f_{S,W}(n) - N >> N^{1-1/r}, the implied constant depending at most on r and c_r; previously known only for r = 2.
• Theorem 2: For W an additive complement of the squares and all large N, sum_{n<=N} f(n) - N >> N^{3/4}/sqrt(T(N)), the implied constant depending only on W, i.e. N^{3/4-o(1)} (Corollary 2), improving the previous N^{1/2}.
• Context (1.1), (1.2): Records the best known lower bound liminf W(N)/sqrt(N) >= 4/pi for squares (Cilleruelo, Habsieger, Balasubramanian-Ramana) and Cilleruelo's Gamma-function bound for r-th powers.