Speaker
Description
The growth of continued-fraction digits is closely related to Diophantine approximation. Products of consecutive digits arise, in particular, in questions concerning improvements to Dirichlet’s theorem. These connections motivate the study of the Hausdorff dimension of sets defined by prescribed growth of digit products.
In this talk, I will discuss weighted products of digits in d-decaying Gauss-like iterated function systems, a class that includes continued fractions and Lüroth expansions. We determine the Hausdorff dimension of sets on which a finite product of digits, raised to arbitrary positive powers and taken at arbitrary fixed positions, exceeds a prescribed function infinitely often. In particular, our result resolves the weighted dimension problem beyond two consecutive digits in the exponential-growth regime, which remained open even for the classical Gauss and Lüroth maps.
A second aspect of the work is that bounded distortion is not assumed. We introduce a weaker property, almost tempered distortion, and use it to prove the existence of geometric pressure defined by cylinder lengths.
This is joint work with Michał Rams.