Ding 2020 green s problem additive complements squares

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Yuchen Ding, Green's problem on additive complements of the squares. Comptes Rendus. Mathématique 358, no. 8 (2020), 897-900. doi:10.5802/crmath.107.

Ding studies Ben Green's question of whether the squares admit an additive complement B = {b_n} with b_n = (pi^2/16)n^2 + o(n^2). Theorem 1 proves that for any additive complement B of the squares, the limsup of ((pi^2/16)n^2 - b_n)/n is at least pi/4, which is far stronger than the earlier Chen-Fang bound of order n^{1/2} log n and confirms their conjecture that the deviation divided by n^{1/2} log n has limsup +infinity. The proof is a short counting argument using only the trivial estimate R(n) >= 1 for the representation function, together with an elementary comparison of the counting function of B against the square-root density; Remark 2 notes the method's simplicity and formulates a further conjecture. For Erdős problem 33 the paper is a citation-trail source: it establishes a second-order obstruction ruling out unusually regular enumerated complements of the squares, but it does not settle Green's question, since a deviation of order n is compatible with b_n = (pi^2/16)n^2 + o(n^2), and it does not improve the limsup-density bounds (the 4/pi-type constants) that problem 33 is actually about.

Source: https://doi.org/10.5802/crmath.107. The file prints "This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/" on its first page: the Creative Commons Attribution 4.0 license.

Bears on. #33

Results to transcribe.

• Theorem 1 (p. 898): If B = {b_n} is an additive complement of the squares, then limsup_n ((pi^2/16)n^2 - b_n)/n >= pi/4, confirming the Chen-Fang conjecture that limsup_n ((pi^2/16)n^2 - b_n)/(n^{1/2} log n) = +infinity.
• Remark 2 (p. 900) / conjecture: The proof uses only R(n) >= 1; the author conjectures that every additive complement B = {b_n} of the squares satisfies limsup_n ((pi^2/16)n^2 - b_n)/n = +infinity.