Hankel determinants and the length of finite Pierce expansions
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…
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