Density Thresholds for Finite-Field Configurations

Definitions and mathematical statement

Let $q$ be a prime power, let $d\geq2$, and let $E\subseteq{\mathbb{F}}_q^d$. Let \[ V=\{0,v_1,\ldots,v_{k-1}\}\subseteq{\mathbb{F}}_q^d \] be the source's nondegenerate $k$-point configuration, with $v_1,\ldots,v_{k-1}$ linearly independent. In particular $k-1\leq d$. The requested copy is specifically \[ \tau+O(V)=\{\tau,\tau+Ov_1,\ldots,\tau+Ov_{k-1}\}\subseteq E, \qquad \tau\in{\mathbb{F}}_q^d,\quad O\in\operatorname{SO}_d({\mathbb{F}}_q). \] The special orthogonal group, not the full orthogonal group or a similarity group, is retained.

For a fixed admissible configuration, an exact threshold formulation is \[ m(q,d,V)=\min\{m:\text{every }E\subseteq{\mathbb{F}}_q^d\text{ with }|E|\geq m \text{ contains some }\tau+O(V)\}. \] Determine this threshold, or its intended sharp order as $q$ grows. If a bound depending only on $(q,d,k)$ is sought, it must hold uniformly over all configurations in the specified nondegenerate class. The record does not specify whether constants may depend on $V$ or the exact precision intended in "how large."

The quadratic form behind $\operatorname{SO}_d$ and the characteristic hypotheses must be fixed. In the usual odd-characteristic standard-form convention, \[ \operatorname{SO}_d({\mathbb{F}}_q)=\{O\in\operatorname{GL}_d({\mathbb{F}}_q):O^{\mathsf T}O=I,\ \det O=1\}. \] This is explanatory, not an added odd-characteristic hypothesis. The record does not exclude characteristic two. Likewise, linear independence of the $v_i$ does not by itself state that their Gram matrix for the intended bilinear form is nonsingular. Whether "nondegenerate" includes that additional quadratic restriction requires source confirmation. No such restriction is guessed.

The supplied literature assessment checked 2026-08-21 labels the problem partially solved. It attributes substantial density thresholds to Hart--Iosevich and later work, for congruent copies or positive proportions of finite-field simplices, while leaving the optimal threshold for the full Problem 5 unresolved. Guaranteeing a positive proportion of congruence classes is not silently identified with containing every prescribed admissible configuration.

Short English statement

Determine the size threshold forcing a translated special-orthogonal copy of a prescribed independent finite-field configuration. Preserve the distinction between fixed-configuration and uniform thresholds, and between affine independence and quadratic nondegeneracy.

Sources

• Local UnsolvedMath registry, permanent ID 30001062, source code catalogue entry; supplied complete record, whose corrected-verified note restores the finite-field configuration notation. The assessment refers to the report's Problem 5. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• Discrete Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/44, pp. 2477--2556: https://doi.org/10.4171/owr/2008/44. Proposed-year metadata gives 2008.
• Chapman--Erdogan--Hart--Iosevich--Koh: https://arxiv.org/abs/0903.4218. The supplied progress assessment also names Hart--Iosevich without a separate bibliographic entry.