On the regularity of the generalised golden ratio function

Given a finite 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 fixed size). Moreover, we demonstrate that as we vary a single parameter m within  A , the generalised golden ratio function may behave like m 1 / hfor 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 ( J. Eur. Math. Soc . 13 (2011) 1113–1146). We provide a new proof of their main result, that is, we explicitly calculate the function G ( { 0 , 1 , m } ) . (For a ternary alphabet, it may be assumed without loss of generality that A = { 0 , 1 , m } with m ∈ ( 1 , 2 ) ] .) We also study the set of m ∈ ( 1 , 2 ] for which G ( { 0 , 1 , m } ) = 1 + m , we prove that this set is uncountable and has Hausdorff dimension  0 . We show that the function mapping m to G ( { 0 , 1 , m } ) 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.