math.NT — Number Theory

Hankel determinants and the length of finite Pierce expansions

Contributed by Jason Davies

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 of each group on curves of bounded degree, whose integer points we count with the uniform bounds of Castryck, Cluckers, Dittmann and Nguyen. The resulting exponent approaches $5/17$ when these curves have large degree relative to the logarithmic size of the box. We raise the degree by adjoining further integer invariants, by an effective Hilbert irreducibility sieve, and by an exact identity for the direction of the curvature vector, and come within $1.61\cdot10^{-5}$ of $5/17$. We also give an elementary proof of the exponent $3/10+\varepsilon$ and a bound for orbits whose quotient increments rarely rise. All results are formalised in Lean 4 with Mathlib. Shallit's conjectured $O((\log a)^2)$ bound remains open.