For 1<β<2 the Hausdorff dimension of the survivor set K(t) equals -lnλ/lnβ, where λ is the smallest positive solution to sum (α_n - t_n) x^n =1, provided the tail sums of the expansion of 1 stay above t.
Sidorov,Supercritical holes for the doubling map, Acta Math
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The-Hausdorff-dimension-of-the-survivor-set
For 1<β<2 the Hausdorff dimension of the survivor set K(t) equals -lnλ/lnβ, where λ is the smallest positive solution to sum (α_n - t_n) x^n =1, provided the tail sums of the expansion of 1 stay above t.