On consecutive 0 digits in the β-expansion of 1

For each 1<β<21<β<2, let rn(β)rn(β) be the maximal length of consecutive 0 digits in the first n digits of 1's β  -expansions. We prove that for Lebesgue almost all 1<β<21<β<2, limn→∞⁡rn(β)/logβ⁡n=1 and rn(β)≥logβ⁡nrn(β)≥logβ⁡n for infinitely many n, which answers two questions raised by Erdös, Joó and Komornik.