Roger Nussbaum – Professor Emeritus, Rutgers University
Date/Time/Location
Thursday, December 4th, 2025, 12:10 pm; Hill Center 705
The Hausdorff Dimension Spectrum of Continued Fractions and Their Generalizations
If T is a nonempty set of of positive integers, denote by dim(T) the Hausdorff dimension of the set of real numbers in [0,1] which can be obtained as a continued fraction using only the elements of T. If S is an infinite set of positive integers we define HD(S), the Hausdorff dimension spectrum of S, to be {s>=0 such that there exists a (possibly infinite) subset T of S with dim(T)=s}. It is known that if S is the set of positive integers, HD(S)=[0,1]. Similar definitions can be given for "complex continued fractions". In general, HD(S) is contained in [0, dim(S)] but may be quite complicated. We shall sketch some of the tools which can be used to study HD(S).