Exceptional digit frequencies and expansions in non-integer bases

Abstract

In this paper we study the set of digit frequencies that are realised by elements of the set of β-expansions. The main result of this paper demonstrates that as β approaches 1, the set of digit frequencies that occur amongst the set of β-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.

Local Dimensions of Overlapping Self-Similar Measures

“…The choice of k varies in the two approaches and depends on ̺. When ̺ < √ 5−1 2 , the k found by Theorem 3.1 results in a tighter upper bound than that found in [1], while for ̺ > √ 5−1 2 , the converse is true. The green curve in Figure 2 shows the upper bound given by Theorem 3.1, while the red curve shows the bound found in [1].…”

“…This is given by the blue curve in Figure 2. Both Theorem 3.1 and Baker in [1], show there exists some k such that…”

“…Proposition 2.2 and Theorem 3.1 give upper bounds on the local dimension of the Bernoulli convolution µ ̺ at x for any x ∈ (0, 1) in the unbiased (̺ > ( √ 5 − 1)/2) and biased (̺ > 1/2) cases respectively. Upper bounds are also given in [1]. In all of these cases, these bounds can be used to show that the local dimension at x = 0 is an isolated point within the set of all possible local dimensions.…”

“…A challenging problem is to find sharp bounds for the set of local dimensions at other x. Upper bounds have recently been found in [1] for special classes of examples of these measures, including the biased Bernoulli convolutions. Our arguments of Section 3 also give upper bounds.…”