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Expected finite k-adic complexity in general integer bases

For an integer k≥3k\ge3, let ENk-adicE_N^{k\text{-adic}} be the expected NNth kk-adic complexity of a uniformly random kk-ary sequence of length N≥1N\ge1. Lin, Xiao and Chen derive a lower bound of the form N/2−O(1)N/2-O(1) and ask, as Question 3.2: estimate an upper bound on ENk-adicE_N^{k\text{-adic}}.

The NNth kk-adic complexity of a sequence is the least log⁡kmax⁡(∣f∣,∣q∣)\log_k\max(|f|,|q|) over reduced rationals f/qf/q whose kk-adic expansion begins with it (unrounded; the convention fixed in Section 1 of the manuscript, following Goresky and Klapper).

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-assisted
Method
Argument
Field
Rational reconstruction; k-adic complexity of finite strings
Posed by
Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.2, AIMS Mathematics 11(7) (2026), p. 20282; DOI 10.3934/math.2026823.
Year posed
2026
Years open
0y
Solved
2026-10
Model
OpenAI GPT models (versions unspecified)
Vendor
OpenAI
Collaborators
Oleksiy Babanskyy
Verification
Unreviewed
Publication
Announced
Significance
4 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The manuscript claims ENk-adic<N/2+0.241933621228105E_N^{k\text{-adic}}<N/2+0.241933621228105 for every integer k≥2k\ge2 and N≥1N\ge1. It gives ENk-adic≤N/2E_N^{k\text{-adic}}\le N/2 for prime-power bases and a strict inequality below N/2N/2 for bases with exactly two distinct prime factors. The example k=30,N=2k=30,N=2 disproves a universal zero-correction bound. Section 1 identifies the standard rational observable with the source question. Written arguments combine classical imports, finite certificates and infinite-tail estimates. The optimal additive constant and finite-length attainment remain open. Constrained FCSR connection integers and rounded register lengths are not identified with this observable. The earlier binary theorem is credited and imported.

What the AI did

As disclosed in the manuscript and AI_USE.md, OpenAI GPT models assisted mathematical exploration, proof criticism, literature searches, manuscript editing, implementation and reproducibility checks. Several AI reviews shared the same project material.Oleksiy Babanskyy retains responsibility for the claims.

Verification

No independent mathematician has checked this yet. Checked by this site on 15 October 2026: the aiXiv v1 record and PDF open (CC BY, content-hashed, not withdrawn), the pinned GitHub commit d673c87 exists, and Question 3.2 was read on printed page 20282 of Lin-Xiao-Chen, AIMS Mathematics 11(7). The manuscript's Theorems 1.2 to 1.4 and its Section 1 comparison of the rational observable with the source's complexity were read. A separate brute force written here computed the exact mean for every base 2 to 60 and every N with base^N up to 70,000 (202 cases): no case reaches N/2 + 0.2419; prime-power bases never exceed N/2; two-prime bases stay strictly below N/2; E for base 30, N = 2 is 1.000256, above 1 as claimed. These are finite checks. The all-base bound rests on the written lattice and tail arguments, Kanold's bound and the finite certificates, none of which were checked or re-run here; the author-side AI reviews are not independent.

Sources

Submitted by Oleksiy Babanskyy on

Changelog2 changes

Discussion