# P2544: Sharp long-shift autocorrelation for half-measure circle sets

- ID: `P2544`
- Reference: `long-shift-autocorrelation-half-set`
- Page: https://theoremdb.org/statements/P2544
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(\mathbb T=\mathbb R/\mathbb Z\) with normalized Lebesgue measure. Determine \(C=\inf_{\mu(A)=1/2}\sup_{1/4\le t\le1/2}\mu(A\cap(A+t))\), where the infimum is over measurable \(A\subseteq\mathbb T\).

### Context

This extremal autocorrelation problem balances a half-measure subset of the circle against all shifts between one quarter and one half.

### Conventions

- **Convention.** The translate \(A+t\) is taken modulo one.
- **Convention.** Sets that differ by a null set are identified.

### What counts as a solution

- Determine \(C\) exactly by giving an exact value or characterization and matching proofs of the upper and lower bounds. Numerical upper and lower bounds alone are partial computational evidence.

## Status

A three-translate argument gives the universal lower bound, while an eight-cell union gives the upper bound. [1](#reference-1)

## Work

### Evidence for the current status

**Proposition 1 (The constant lies between 1/6 and 3/16).** A three-translate argument gives the universal lower bound, while an eight-cell union gives the upper bound.

For
\[
R_A(t)=\mu(A\cap(A+t)),
\qquad
M(A)=\sup_{1/4\leq t\leq1/2}R_A(t),
\]
the two certified bounds are
\[
\frac16\leq C=\inf_{\mu(A)=1/2}M(A)\leq\frac3{16}.
\]
The lower bound follows already at \(t=1/3\). The upper bound is attained by the displayed 16-cell set in `lshac-claim-sixteen-cell-upper-bound`. Their gap is
\[
\frac3{16}-\frac16=\frac1{48}.
\]
This falls short of the candidate's requested \(10^{-8}\) gap. The exact value remains unresolved in this entry.

### Background and intake notes

- Original intake status: Status not established. No literature search was performed. Popular-difference inequalities and covariograms may contain this extremal constant.
- Averaging the autocorrelation over the whole circle gives 1/4, but the target shift interval excludes a neighborhood of zero. That average does not give the desired lower bound.
- Discretization needs a rounding theorem or compactness argument before finite cyclic lower bounds transfer to measurable sets.

- Recorded example: Partition the circle into 16 equal half-open cells and take cell indices \(\{0,1,2,3,4,5,7,10\}\). This gives \(C\le3/16\).

### Other known results

- **Theorem 1** (established): Counting pairs among three one-third translates proves R_A(1/3) >= 1/6. [1](#reference-1)
- **Computation 1** (reproduced): The union of cells 0,1,2,3,4,5,7,10 has boundary-shift intersection counts 3,3,3,3,2. [1](#reference-1)

### Prior approaches

- **Route 1** (inconclusive): The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts. [1](#reference-1) [3](#reference-3) [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python verifies the continuous 3/16 construction and exhausts five finite cyclic half-set problems. [1](#reference-1)

### Computational notes

- For the stated 16-cell set, exact cyclic intersection counts at shifts 4,5,6,7,8 are 3,3,3,3,2. Piecewise linearity between cell-boundary shifts proves the continuous supremum is 3/16. Exhaustive half-subset enumeration on cyclic grids of sizes 4,8,12,16,20 gave discrete optima 1/4,1/4,1/4,3/16,1/5 for the corresponding grid-shift problem; these finite optima are not continuous lower bounds.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `long-shift-autocorrelation-half-set`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Gennadiy Averkov and Gabriele Bianchi, “Confirmation of Matheron's conjecture on the covariogram of a planar convex body”. arXiv:0711.0572 (2007). Gennadiy Averkov and Gabriele Bianchi, Confirmation of Matheron's conjecture on the covariogram of a planar convex body, Journal of the European Mathematical Society 11 (2009), 1187-1202, definition and introduction; popular-difference search performed 2026-07-24 https://arxiv.org/abs/0711.0572
   - Also cited at Independent proof and exact construction recorded in the two supporting claims, 2026-07-24
   - Also cited at Independent three-translate counting argument, 2026-07-24
   - Also cited at Explicit construction and exact replay in lshac-artifact-cell-and-grid-enumeration
   - Also cited at Inline CPython standard-library computation executed on 2026-07-24
   - preprint; primary source; arXiv:0711.0572, version checked 2026-07-24; checked 2026-08-01
   - Source use: original_summary
   - For Sharp long-shift autocorrelation for half-measure circle sets: The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts.
   - Source named by the research packet.
2. <a id="reference-2"></a>Aaron Berger, “Popular Differences for Corners in Abelian Groups”. Math. Proc. Camb. Phil. Soc. 171 (2021) 207-225. DOI 10.1017/S0305004120000195. arXiv:1909.12350 (2019). Popular Differences for Corners in Abelian Groups, compact-abelian-group setting and abstract https://arxiv.org/abs/1909.12350
   - preprint; reference source; arXiv:1909.12350, version checked 2026-07-24; checked 2026-07-24
   - Source use: citation_only
   - For Sharp long-shift autocorrelation for half-measure circle sets: The located covariogram literature does not give this interval extremum. The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts.
3. <a id="reference-3"></a>Gabriele Bianchi, “Some Open Problems Regarding the Determination of a Set from Its Covariogram,” Le Matematiche 60(2) (2005), 247-257. Gabriele Bianchi, Some open problems regarding the determination of a set from its covariogram, Le Matematiche 60 (2005), 247-257 https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/148
   - website; reference source; web version checked 2026-07-24; checked 2026-07-24
   - Source use: citation_only
   - For Sharp long-shift autocorrelation for half-measure circle sets: The located covariogram literature does not give this interval extremum. The search found the standard covariogram framework and popular-difference results, with no sharp theorem for the fixed arc of shifts.