# P2880: Zero density in the easily bored sequence

- ID: `P2880`
- Reference: `easily-bored-sequence-zero-density`
- Page: https://theoremdb.org/statements/P2880
- Record maturity: Reviewed problem with recorded work

## Problem

For a finite binary word \(w\) and \(x\in\{0,1\}\), let \(r_w(x)\) be the largest integer \(r\ge 1\) for which \(wx\) ends in \(u^r\) for some nonempty word \(u\), and let \(\ell_w(x)\) be the largest length \(|u|\) among witnesses for that maximal \(r\). Define \(b_1=0\); for \(m\ge 2\), let \(b_m\) be the bit \(x\) that minimizes \((r_{b_1\cdots b_{m-1}}(x),\ell_{b_1\cdots b_{m-1}}(x))\) in lexicographic order. Does the limit \(\lim_{N\to\infty}N^{-1}|\{1\le m\le N:b_m=0\}|\) exist and equal \(1/2\)?

### Remarks

- **Remark.** For a word u, the power u^r is the concatenation of r copies of u.
- **Remark.** Lexicographic minimization first minimizes the number of repetitions r and then, when those agree, minimizes the length of the repeated suffix block.

### What counts as a solution

- Prove that the displayed limit exists and equals 1/2, or prove that it fails to exist or has a different value.
- Any computation offered as evidence must publish an exact generator, prefix length, counts of both bits, and a digest of the generated prefix.

## Status

UNKNOWN as of 2026-07-31. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution. Prove that the displayed limit exists and equals 1/2, or prove that it fails to exist or has a different value. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution. Prove that the displayed limit exists and equals 1/2, or prove that it fails to exist or has a different value.

UNKNOWN as of 2026-07-31. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution.

A complete resolution must satisfy this condition: Prove that the displayed limit exists and equals 1/2, or prove that it fails to exist or has a different value.

### Background and intake notes

This deterministic word is inexpensive to extend but difficult to analyze. Prefix counts, first occurrences of large powers, and automaton-like recurrence states are useful shared data when tied to exact generator versions.

- Original intake status: UNKNOWN as of 2026-07-27. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution.
- On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 377105; its comments give computations rather than a proof.
- Della Corte, Topology and its Applications 320 (2022), 108244, studies this exact sequence and lists zero frequency 1/2 as Conjecture 1 while proving other dynamical facts.
- OEIS A337546 links the paper and continues to label density 1/2 as conjectural; a search of the paper title and its citations located no later proof.
- A TheoremDB search for the easily bored sequence, A337546, repetition-pair recurrence, and zero density found no duplicate.

- Recorded example: The sequence begins 0,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0,1,1,0.

### Open directions

- **Route 1** (reported): Prove that the displayed limit exists and equals 1/2, or prove that it fails to exist or has a different value. [1](#reference-1)

### Computational notes

- The 2022 paper reports zero frequency 0.5001 in the first 50,000 digits, which supports the target without settling convergence.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `easily-bored-sequence-zero-density`, 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>MathOverflow: The easily bored sequence. Question 377105 and all visible comments, checked through the Stack Exchange API on 2026-07-27. Question 377105 and all visible comments, checked through the Stack Exchange API on 2026-07-27. https://mathoverflow.net/questions/377105/the-easily-bored-sequence
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Zero density in the easily bored sequence: UNKNOWN as of 2026-07-27. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution.
   - Source named by the research packet.
2. <a id="reference-2"></a>Alessandro Della Corte, “The Easily Bored Sequence”. Topology and its Applications 320 (2022), 108244. DOI 10.1016/j.topol.2022.108244. Full journal article relevant to Zero density in the easily bored sequence. https://doi.org/10.1016/j.topol.2022.108244
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Zero density in the easily bored sequence: UNKNOWN as of 2026-07-27. The source page has zero answers. A 2022 peer-reviewed paper on the sequence lists existence and value 1/2 of the zero frequency as its first conjecture, and the checked databases show no later resolution.
3. <a id="reference-3"></a>Alessandro Della Corte, “A337546: A binary sequence defined by minimizing final consecutive repeated words,” On-Line Encyclopedia of Integer Sequences, submitted November 22, 2020, checked 2026-08-01. definition, comments, conjecture 2, first 10,000 terms, and links https://oeis.org/A337546
   - reference_database; reference source; checked 2026-07-31
   - Source use: citation_only
   - Reused material: definition, comments, conjecture 2, first 10,000 terms, and links
   - Reuse basis: fair_use_reviewed; rights holder: The OEIS Foundation Inc. and the credited contributors; checked 2026-08-01; by Philip Weiss, TheoremDB staff
   - Required attribution: Alessandro Della Corte, “A337546: A binary sequence defined by minimizing final consecutive repeated words,” On-Line Encyclopedia of Integer Sequences, submitted November 22, 2020, checked 2026-08-01.
   - Source used to assess the problem's recorded status.
   - For Zero density in the easily bored sequence, this source defines the easily bored sequence and records the conjecture that the zero density exists and equals one half.