Bifurcations of digit frequencies in unique expansions.

For m ∈ N and β ∈ (1 , m + 1 ] , we consider the set U m , β consisting of unique β -expansions in { 0 , 1 , ⋯ , m } N ∖ { 0 ∞ , m ∞ }. Let k ∈ { 1 , ⋯ , m } with k > k ‾ where k ‾ : = m − k. We determine the bifurcation value of β 's, below which in any w ∈ U m , β the digit frequencies of k and k ‾ exist and are equal, and above which there are many w ∈ U m , β , consisting of a set of positive dimension, such that the digit frequencies of k and k ‾ in w do not exist. We also determine the bifurcation value of β 's, below which in any w ∈ U m , β the upper and lower frequencies of k are respectively equal to the upper and lower frequencies of k ‾ , and above which there exists c > 0 , such that for any r ∈ (− c , c) , there are many w ∈ U m , β , consisting of a set of positive dimension, such that the difference of the digit frequencies of k and k ‾ in w is exactly equal to r. For a video summary of this paper, please visit https://youtu.be/IJvlWTt5DIQ. [ABSTRACT FROM AUTHOR]