Doorn 2025 smallest set such that every positive

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limsup_sharpness: Proves that the explicit square complement attains the stated limsup constant, without claiming this constant is optimal among complements.

main_theorem: Constructs a complement of the squares with counting function strictly below twice the golden ratio to the power five halves times sqrt(x).

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Wouter van Doorn, The smallest set such that every positive integer is the sum of a square and an element from this set. Two-page author note, 2025, whose reference [2] is marked accessed 03-10-2025. The note has one unnumbered Theorem (p. 1, proof pp. 1–2) and one unlabelled sharpness remark (p. 2).

Source and version

The edition read is the author's pinned public PDF, identified by its file hash. No notice is printed in the two-page note, and the author's source repository has no license file and no license statement in its file list (https://github.com/Woett/Mathematical-shorts, read 2026-10-02); the term is unstated.

The author posted the construction in the #33 discussion, with an updated PDF linked on 2025-10-03. The main problem page attributes the bound to van Doorn. This records public attribution and adoption; no journal publication or exhaustive priority determination is asserted.

Results and method

The main theorem constructs a positive-integer set $A$ such that every positive integer is $a+n^2$ with $a\in A$ and $n\geq0$, and

$$A(x)<2\varphi^{5/2}\sqrt{x}\qquad(x>0), \qquad \varphi=\frac{1+\sqrt5}{2}.$$

The construction places short integer windows near the geometrically spaced points $\varphi^{2j}$. Rounding a square root puts every integer into one of those windows after subtracting a square. A geometric sum controls the total number of selected integers.

The sharpness remark asserts, with only the hint to take $x=\varphi^{2j}+2\varphi^{j+1/2}-1$ for large $j$, that this particular set has limsup exactly $2\varphi^{5/2}$, approximately $6.66$. The theorem's page gives a proof sketch written here, noting that one intermediate inequality of the print is non-strict when the fractional part is zero, which leaves the final strict bound intact. The remark's page gives a proof written here, including disjointness of the windows and integer rounding.

Read status. Claims checked for the Theorem (p. 1) and the remark (p. 2), against the print; the arguments on both result pages were checked step by step. Nothing is independently reviewed.

Relation to Problem 33

The construction supplies a public upper bound for the infimum in #33, including the stronger requirement that every positive integer is represented. It does not determine the infimum and does not contradict the lower bound $4/\pi$ for the liminf recorded on the problem page.

The representation-excess estimates of Ding, Krause, Sándor, Sun, and Zhang measure a different aspect of square complements: the surplus of representations, rather than the optimal limsup counting constant. Their current preprint does not determine that constant either.

Bears on.

• #33: the Theorem shows the smallest possible limsup asked for is at most $2\varphi^{5/2}\approx6.66$, by a set representing every positive integer; the remark shows this set's limsup equals that constant. An upper bound only; the problem's value is not determined.

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