Descendants of the mother of all continued fractions

Slade Sanderson (Utrecht U)

Apr 23. 2025, 14:00 — 14:45

In 1855, Seidel introduced an arithmetic procedure, called contraction, which produces from a given generalised continued fraction (GCF) a new GCF whose convergents are any prescribed subsequence of the original GCF-convergents.  In 1989, Shunji Ito gave a planar natural extension of the Farey tent map, which generates `slow' GCF-expansions (Farey expansions) and which has—up to isomorphism—been called `the mother of all continued fractions.'  In this talk, we `induce contractions of the mother of all continued fractions' (formally, we use induced transformations of Ito's natural extension to govern contractions of Farey expansions) and highlight some of the GCF-algorithms that are born from this procedure.  Within our setting, we find several well-studied expamples, including regular continued fractions, Kraaikamp's S-expansions, and Nakada's α-continued fractions for all α between 0 and 1, and we introduce new, superoptimal continued fractions which simultaneously converge arbitrarily fast and possess arbitrarily good approximation properties.  This is joint work with Karma Dajani and Cor Kraaikamp.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Files:
Slides
Associated Event:
Uniform Distribution of Sequences (Workshop)
Organizer(s):
Henk Bruin (U of Vienna)
Robbert Fokkink (TU Delft)
Jörg Thuswaldner (Montanuniversität Leoben)