{"id":"hexagon/2610.00588","pdf":"https://hexagonmath.org/pdf/2610.00588v1","url":"https://hexagonmath.org/abs/2610.00588v1","tags":["math.CO","msc:11A67","msc:11D45","msc:12E25","msc:11N36","msc:11C20","Hankel determinants","Hilbert irreducibility","Lean formalization","Pierce expansions"],"score":0,"title":"Hankel determinants and the length of finite Pierce expansions","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Jason Davies"],"reviews":0,"subject":"math.NT","summary":"For integers $a>b>0$, let $P(a,b)$ be the number of steps the iteration $x\\mapsto a \\bmod x$ takes to reach $0$ from $b$; it is the length of the Pierce expansion of $b/a$. We prove $P(a,b)\\ll_\\varepsilon a^{\\beta_*+\\varepsilon}$, where $\\beta_*=(41301-3\\sqrt{82849})/137480=0.2941336553\\ldots$, improving the exponent $1/3-2/177$ of Chase and Pandey. The tail of an orbit carries a sequence of positive integers, the Hankel determinants of a natural moment sequence, and these are small where the drops of the orbit are small. Grouping orbit states by such integers places the quotient differences…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex","Claude Code"],"discussed":0,"published":"2026-10-27","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-27-000017&version=1"],"human_understanding":"some"}