[#P2880] Zero density in the easily bored sequence
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 \(1/2\)?
1Context
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.
2Problem setup
Definition 1 (For a word u, the power u^r). For a word u, the power u^r is the concatenation of r copies of u.
Definition 2 (Lexicographic minimization first minimizes the number of repetitions r and then, when those agree, minimizes the length of the repeated suffix block). Lexicographic minimization first minimizes the number of repetitions r and then, when those agree, minimizes the length of the repeated suffix block.
Remark 1. 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.
3What 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.
1Status
Current status (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.[1]
1Records
Notes and companion material
Original intake 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.
- 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 1. 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.
Computational notes
- The 2022 paper reports zero frequency 0.5001 in the first 50,000 digits, which supports the target without settling convergence.
How the 2 records connect
ProblemZero density in the easily bored sequence
2See also
- Minimum avoiding alphabet for every avoidable wordcombinatorics on words
- Infinitely many ones in the greedy three-term-progression-free sequencecombinatorics on words
- Least uniform modulus of an abelian-square-free morphism on four letterscombinatorics on words
How to cite
TheoremDB contributors, “Zero density in the easily bored sequence,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/easily-bored-sequence-zero-densityThis page as plain text: easily-bored-sequence-zero-density.md
This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗forum · discovery source · checked 2026-07-31Source use: original summary.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.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.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.
- Alessandro Della Corte, “The Easily Bored Sequence”. Topology and its Applications 320 (2022), 108244. DOI 10.1016/j.topol.2022.108244. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.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.Also cited at Full journal article relevant to Zero density in the easily bored sequence.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.
- 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. Status evidence identified in the source record and checked at the linked publication. ↗website · primary source · checked 2026-07-31Source use: original summary.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.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.Also cited at definition, comments, conjecture 2, first 10,000 terms, and links.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.
This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.