GPT 6 Astra Ultra attempt on Math-5668

Erdős Problem 465

Statement and scope.

Let $N(X,\delta)$ be the largest number of points in a disc of radius $X$ whose pairwise distances have distance at least $\delta$ from every integer. For fixed $0<\delta<1/2$, must $N(X,\delta)=o(X)$, and even $N(X,\delta)\leq X^{1/2+o(1)}$? Both conclusions are known; Konyagin proved $O_\delta(\sqrt X)$. The sublinear step is unresolved here. We prove a uniform linear bound and a sharper bound on the collinear special case.

A strip and fractional-coordinate bound.

Translate the disc to the origin. Divide the vertical interval $[-X,X]$ into at most $\lceil8X/\delta\rceil+1$ half-open strips of height at most $\delta/4$. Divide the fractional-coordinate circle, represented by $[0,1)$, into $\lceil4/\delta\rceil$ half-open intervals of length at most $\delta/4$. Assign each point $(x,y)$ the strip containing $y$ and the interval containing $\{x\}$.

Two different points with the same assignment would satisfy \[ |\Delta y|\leq\delta/4,\qquad \||\Delta x|\|\leq\delta/4. \] The triangle inequality gives \[ 0\leq\sqrt{\Delta x^2+\Delta y^2}-|\Delta x| \leq|\Delta y|. \] The distance-to-integers function is $1$-Lipschitz, so their Euclidean distance would have distance at most $\delta/2<\delta$ from an integer. This contradicts the hypothesis. Thus every assignment contains at most one point, and \[ N(X,\delta)\leq \left(\lceil8X/\delta\rceil+1\right)\lceil4/\delta\rceil =O_\delta(X). \tag{1} \] This improves the elementary area-packing bound of order $X^2$.

The collinear case.

For points on a line, choose signed coordinates $t_i$ along it. Then $\||P_i-P_j|\|=\|t_i-t_j\|$. Their residues modulo one are therefore pairwise separated by circular distance at least $\delta$. Ordering these residues around the circle, every gap is at least $\delta$, and their sum is one. There can be at most $\lfloor1/\delta\rfloor$ points. In particular, unbounded constructions require variation in both coordinates.

Remaining gap and sources.

The assignments in $1$ impose no useful restriction between different vertical strips. Obtaining a vanishing fraction of occupied assignments, or the square-root bound, requires additional geometric information. The known result is S. V. Konyagin, On the distances between points on the plane, Mathematical Notes 69 $2001$, 578--581, https://www.mathnet.ru/eng/mzm691. Current statement and Sárközy's earlier sublinear estimate: https://www.erdosproblems.com/465, checked 24 September 2026. Neither external upper-bound theorem is used to fill the gap in this attempt.

Completion estimate.

The linear bound and collinear case are proved; the main sublinear improvement is missing. COMPLETION: 30%