Circle Discrepancy in Balanced Lattice Colorings

Definitions and mathematical statement

The record aggregates four historical circle-discrepancy questions: obtain unbounded discrepancy for translates of circles in globally balanced lattice colorings; prescribe the radius if possible; study one-sided discrepancy; and find further natural shapes with analogous behavior. Its supplied verification marks the extraction unrecoverable as one problem, says the report immediately discusses historical questions as answered or superseded, and places the remaining quantitative fixed-radius conjecture in a separate record. The aggregate is not turned into four new assertions of open status here.

A precise model must first specify the coloring domain and discrepancy. For example, lattice-point discrepancy sums colors of points in a chosen set, whereas a unit-square coloring of the plane could be integrated over a circle boundary or over its enclosed disk. These are different quantities. The source does not say which circle convention, counting/length/area measure or boundary rule is intended. Nor does it define globally balanced: exact balance on a finite board and zero limiting mean on an infinite lattice are not identical conditions.

Once a signed discrepancy $\Delta_\chi(c,r)$ is fixed for center $c$ and radius $r$, define \[ D_\chi(r)=\sup_c|\Delta_\chi(c,r)|, \quad D_\chi^+(r)=\sup_c\Delta_\chi(c,r), \quad D_\chi^-(r)=\sup_c(-\Delta_\chi(c,r)). \] The historical questions can then be distinguished as growth of $D_\chi$ while radii vary, lower bounds at a prescribed radius, and bounds for one or both signed quantities. A large absolute discrepancy does not specify its sign. If the board is finite, unboundedness additionally requires a varying-size family or another stated asymptotic parameter; it cannot mean exceeding every bound on one fixed finite set.

The literature triage checked 21 August 2026 reports power-type lower bounds when a radius can be chosen from a pair of scales, but says fixed-radius and one-sided quantitative variants are not fully settled. The actual scales, exponent, coloring hypotheses and remaining separate conjecture are not supplied. A result allowing either of two radii is not silently replaced by a result for every prescribed radius. The historical answered/superseded report and this more qualified remaining-gap assessment are both retained, without guessing their precise reconciliation or inventing the missing quantitative conjecture.

Short English statement

This is an aggregate of historical questions with an unrecoverable exact model. Circle boundaries, disks, lattice counts and planar integrals must be distinguished. The dated review reports two-scale lower bounds, while prescribed-radius and signed quantitative questions require their separate precise statements.

Sources

• Local UnsolvedMath registry, permanent ID 30000049, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026; statement marked unrecoverable.
• 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 reference "Circle discrepancy": https://arxiv.org/abs/1201.5544.
• Supplied reference "OWR 13/2004": https://doi.org/10.4171/OWR/2004/13.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. Full theorem and separate fixed-radius conjecture statements are not supplied.