The run-length function of the β-expansion of the unit

For any real number β>1, the run-length function rn(β) is defined as the maximal length of consecutive zero digits amongst the first n digits in the β-expansion of 1. It was known that rn(β) grows to infinity with the speed logβ⁡n for Lebesgue almost all β∈(1,2). In this note, we quantify the size of the set of β for which rn(β) grows to infinity in a general speed. More precisely, for any strictly increasing function ϕ:N→R+ with ϕ(n) tending to +∞ and ϕ(n)/n decreasing to 0 as n→∞, we prove that for any real numbers 0≤a≤b≤+∞, the setEa,b={β∈(1,2):liminfn→∞rn(β)ϕ(n)=a,limsupn→∞rn(β)ϕ(n)=b} is of full Hausdorff dimension.