Counting Lattice Oval Configurations by Weight

Definitions and mathematical statement

Let $\Lambda\subseteq{\mathbb{R}}^2$ be a fixed full-rank planar lattice and $S\subseteq{\mathbb{R}}^2$ a fixed closed bounded convex region containing the origin. Under the usual homothetic-oval convention, put \[ S(r,P)=P+rS,\qquad r>0,\quad P\in{\mathbb{R}}^2, \] \[ J(r,P)=\Lambda\cap S(r,P),\qquad N(r,P)=|J(r,P)|. \] The record refers to earlier "same lattice and $S$-oval notation" without explicitly defining $S(r,P)$, so this homothety/translation interpretation requires confirmation. Closedness means that points on the boundary are included. No rotation of $S$ is introduced.

Two configurations, viewed as finite subsets of $\Lambda$, are equivalent precisely when \[ J\sim J'\quad\Longleftrightarrow\quad J'=J+v\text{ for some }v\in\Lambda. \] Only distinct subsets modulo lattice translation are counted, not parameter pairs $(r,P)$, which can yield the same subset. For a nonnegative integer $n$, define \[ \mathcal C_n(\Lambda,S)= \{J(r,P):r>0,\ P\in{\mathbb{R}}^2,\ N(r,P)=n\}/\!\sim, \qquad K(n)=|\mathcal C_n(\Lambda,S)|. \] For a nonnegative integer $N$, put \[ M(N)=\left|\{J(r,P):r>0,\ P\in{\mathbb{R}}^2,\ N(r,P)\leq N\}/\!\sim\right| =\sum_{n=0}^N K(n), \] when the empty configuration is included; a positive-weight convention omits $n=0$. The questions ask to determine these counts as functions of the weight and cutoff for the fixed lattice and shape, including their intended growth/asymptotic behavior. No specific proposed formula, exponent, leading constant, or error term is supplied.

The original says translation by an integer vector, while the clean statement says a lattice vector. These agree after choosing a lattice basis identifying $\Lambda$ with ${\mathbb{Z}}^2$; they need not agree for literal ambient integer vectors without that identification.

The supplied assessment dated 2026-08-21 reports sharp bounds by Huxley--Zunic for quantities called $L(N)$ and $M(N)$ under a Quadrangle Condition, while general and circular cases retain unresolved asymptotic questions. Neither that condition nor $L(N)$ is defined here, so $L$ is not identified with the original $K$, and the conditional bounds are not transferred to arbitrary $S$.

Short English statement

Count the different lattice-point subsets cut out by translated, scaled copies of one fixed convex shape, identifying subsets that differ by a lattice translation. The problem asks both for a fixed number of points and for at most a given number, rather than counting the continuously varying placement parameters.

Sources

• Local UnsolvedMath registry, permanent ID $30000938$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Analytic Number Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/14, pp. 669--746. https://doi.org/10.4171/owr/2008/14.
• Supplied reference: Huxley and Zunic, "configurations II," https://doi.org/10.1112/plms/pdt011.
• The supplied corrected_verified note says the full statement was retained with notation placed in delimiters. The partially solved label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of the Quadrangle Condition, definitions, bounds, and remaining asymptotic scope.