Sothanaphan 2026 compact poissonbessel proof integer distance free

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lemma_1_1: For 0 < s < 1 the kernel K_s(t), the sum over k at least 1 of (k + 2 s k^2) e^(-sk) J_0(2 pi k t), is positive definite on the plane as a function of |x|, satisfies |K_s(t)| <= K_s(0) << s^(-2), and equals an absolutely convergent sum over integers m of explicit terms T_m(s,t).

lemma_1_2: There are constants L >= 1 and s_0 > 0 such that, for 0 < s < s_0 and m >= 1, each term T_m(s,t) of the Poisson-Bessel expansion is non-positive once |2 pi t - 2 pi m| >= L s, with explicit negative bounds on each side of 2 pi m, and, after increasing L, T_0(s,t) <= 0 once 2 pi t >= L s.

proposition_1_3: There are constants B, c, s_0 > 0 such that for 0 < s < s_0 the Poisson-Bessel kernel satisfies K_s(t) <= -c (1+t)^(-1/2) whenever the distance from t to the nearest integer is at least B s.

theorem_2_1: The manuscript's main theorem: for R at least 1, a measurable subset of the planar disc of radius R with no two distinct points at a positive integer distance has measure at most a constant times R^(1/2); with Sárközy's construction, M(R) = R^(1/2+o(1)).

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Nat Sothanaphan, A compact Poisson–Bessel proof for integer-distance-free planar sets. manuscript (2026).

The manuscript is dated 29 April 2026 and runs to four pages. Sothanaphan records a streamlined version of Chojecki's recent argument on Erdős Problem 953: writing $M(R)$ for the supremum of the measures of measurable subsets of the disc $B_R(0)$ of the plane with no two distinct points at a positive integer distance (p. 1), Theorem 2.1 (p. 4) proves $M(R)\ll R^{1/2}$ for $R\ge1$, which with Sárközy's construction, cited for the lower bound, gives $M(R)=R^{1/2+o(1)}$ (pp. 1, 4). The method is a Delsarte-type energy bound with a radial kernel expanded in Bessel functions $J_0$: for $0<s<1$, Lemma 1.1 (p. 2) takes $K_s(t)=\sum_{k\ge1}(k+2sk^2)e^{-sk}J_0(2\pi kt)$, obtained from the Abel-smoothed sum $\sum_{k\ge1}e^{-sk}J_0(2\pi kt)$, shows that $x\mapsto K_s(|x|)$ is positive definite on the plane with $|K_s(t)|\le K_s(0)\ll s^{-2}$, and expands it by Poisson summation into terms $T_m(s,t)$ indexed by the integers $m$. Lemma 1.2 (p. 2) signs those terms, and Proposition 1.3 (p. 3) gives the key negativity: there are $B,c,s_0>0$ with $K_s(t)\le-c(1+t)^{-1/2}$ whenever $0<s<s_0$ and $\|t\|_{\mathbb Z}\ge Bs$. The energy inequality then sets an $O(m)$ near-diagonal contribution, for a compact subset of measure $m$, against a negative off-diagonal contribution of size about $R^{-1/2}m^2$, forcing $m\ll R^{1/2}$. The manuscript states that it was produced with use of GPT-5.5 Thinking.

Source: https://drive.google.com/file/d/1jthm5EkUg5l8nnSCB0Ojk0YJteJP6L9P/view. The copy read for this card is the author's four-page manuscript from that Google Drive share, which prints no notice; the share states no license for the file, and an arXiv author query on 2026-10-02 found no arXiv record for the paper; the term is unstated.

Read status: claims checked for Lemmas 1.1 and 1.2, Proposition 1.3 and Theorem 2.1, read clause by clause on the page images of the manuscript; the proofs were followed at the level recorded on each result page, and Sárközy's construction was not read here. Nothing here is independently reviewed by this corpus; the outside review of the argument is recorded on the problem's claim page.

Bears on. #953: Theorem 2.1 gives $M(R)\ll R^{1/2}$ for $R\ge1$ and, with Sárközy's cited lower bound, the exponent in $M(R)=R^{1/2+o(1)}$; it does not decide whether $M(R)$ has order exactly $R^{1/2}$, which the problem page leaves open. The bound is Chojecki's, recorded at his Theorem 1.1; this manuscript gives a shorter proof of it.

Results. Labels and pages are those of the manuscript named above (pp. 1-4).

• Lemma 1.1 (p. 2): for $0<s<1$, $x\mapsto K_s(|x|)$ is positive definite on $\mathbb R^2$, $|K_s(t)|\le K_s(0)\ll s^{-2}$, and $K_s(t)=\sum_{m\in\mathbb Z}T_m(s,t)$, absolutely convergent, with $T_{-m}=T_m$ given explicitly.
• Lemma 1.2 (p. 2): there are $L\ge1$ and $s_0>0$ such that for $0<s<s_0$, $m\ge1$ and $|2\pi m-2\pi t|\ge Ls$, the term $T_m(s,t)$ satisfies an explicit negative bound when $2\pi t<2\pi m$ and is non-positive when $2\pi t>2\pi m$; after increasing $L$, $T_0(s,t)\le0$ whenever $2\pi t\ge Ls$.
• Proposition 1.3 (p. 3): there are $B,c,s_0>0$ with $K_s(t)\le-c(1+t)^{-1/2}$ whenever $0<s<s_0$ and $\|t\|_{\mathbb Z}\ge Bs$.
• Theorem 2.1 (p. 4): $M(R)\ll R^{1/2}$ for $R\ge1$; with Sárközy's construction, $M(R)=R^{1/2+o(1)}$.

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