On the regularity of the generalised golden ratio function

Abstract

Given a nite set of real numbers A, the generalised golden ratio is the unique real number G(A) > 1 for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that G(A) varies continuously with the alphabet A (of xed size). What is more, we demonstrate that as we vary a single parameter m within A, the generalised golden ratio function may behave like m 1=h for any positive integer h. These results follow from a detailed study of G(A) for ternary alphabets, building upon the work of Komornik, Lai, and Pedicini (2011). We give a new proof of their main result, that is we explicitly calculate the function G(f0; 1;mg). (For a ternary alphabet, it may be assumed without loss of generality that A = f0; 1;mg with m 2 (1; 2)].) We also study the set of m 2 (1; 2] for which G(f0; 1;mg) = 1 + p m; we prove that this set is uncountable and has Hausdor dimension 0. We show that the function mapping m to G(f0; 1;mg) is of bounded variation yet has unbounded derivative. Finally, we show that it is possible to have unique expansions as well as points with precisely two expansions at the generalised golden ratio.