Power-Law Fixed-Radius Circle Discrepancy

Definitions and mathematical statement

The source asks for a power lower bound at every prescribed radius, rather than after choosing between two radii. Write $T_n={\mathbb{R}}^2/n{\mathbb{Z}}^2$ and let $\chi$ be a red-blue coloring of its $n\times n$ toroidal square lattice, encoded by $\pm1$, with global balance $\sum\chi=0$. For a point or unit-cell coloring this requires $n$ even. If $\Delta_\chi(t,R)$ denotes the intended absolute color imbalance of the translated circle centered at $t$, the proposed quantifiers are \[ \exists c>0\ \exists\alpha>0\ \forall\text{ even }n\ \forall\chi\text{ balanced}\ \forall R\in(2,n/2)\ \exists t\in T_n:\quad \Delta_\chi(t,R)\geq cR^\alpha. \] The constants must be independent of $n$, the coloring and the prescribed radius; the center may depend on all of them. If the radius interval is empty the assertion is vacuous.

The record does not define "circle imbalance" precisely. In the common unit-cell boundary-length model, extend $\chi$ to a periodic function constant on each unit square and set \[ \Delta_\chi(t,R)=\left|\int_{C(t,R)}\chi(x)\,ds(x)\right| =\left|R\int_0^{2\pi}\chi\left(t+R(\cos\theta,\sin\theta)\right)\,d\theta\right|. \] This measures red versus blue arc length. In a lattice-vertex model a filled-disk discrepancy would instead be the absolute signed count of vertices in $B_{T_n}(t,R)$; signed counts on the circumference and integrals over a filled disk are still different quantities. The extraction's "square lattice" and "circle" do not resolve these choices. Thus the arc-length formula is an explicit possible convention, not a silent identification of the source model. The restriction $R<n/2$ permits an embedded Euclidean circle or disk on the flat torus.

The supplied literature assessment checked 21 August 2026 reports a logarithmic lower bound and a later power bound only when a radius or its double may be used. It did not locate a power bound for each prescribed radius. In particular, a statement of the form \[ \max\{\sup_t\Delta_\chi(t,R),\sup_t\Delta_\chi(t,2R)\} \geq cR^\alpha \] in its applicable model and parameter range does not imply the fixed-$R$ assertion above. The assessment does not specify the logarithmic rate, power exponent, or the doubled-radius range restrictions, so those are not invented here.

Short English statement

Must every balanced toroidal coloring have a translated circle with polynomially large imbalance at each prescribed radius? The center can vary, but the radius cannot. The supplied assessment distinguishes the open fixed-radius question from a power bound allowing either a radius or its double; the precise coloring and circle-measure convention still needs source review.

Sources

• Local UnsolvedMath registry, permanent ID 30000050, exact source code catalogue entry; Power-Law Fixed-Radius Circle Discrepancy. Proposed year 2004, status open; corrected_verified wording specifies absolute $c,\alpha>0$ and $2<R<n/2$.
• Discrepancy Theory and Its Applications (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/13, pp. 673--722, https://doi.org/10.4171/owr/2004/13.
• Supplied literature triage checked 2026-08-21: "Circle discrepancy", https://arxiv.org/abs/1201.5544; "OWR 13/2004", https://doi.org/10.4171/OWR/2004/13. This dated OWR-derived curation requires independent verification of time-sensitive status and cited claims.