This paper computes the Hausdorff dimension of a subset of [0,1] defined by a particular restriction of the expansion in some integer base b. Specifically, let \(T_ b(c,r)\) be the set of \(x\in (0,1)\) such that in the base-b expansion of x, every r consecutive digits have sum at least c. The authors show \[ \dim T_ b(c,r)=[\log \rho (M)]/\log b \] where M is a 0-1 matrix they construct, and \(\rho\) (M) is its spectral radius.
The authors give a direct proof. Their result is, however, a special case of a result of H. Furstenberg [Math. Systems Theory 1, 1-49 (1967; Zbl 0146.285)] which shows that for any compact set \(K\subset [0,1]\) which is invariant under \(\tau_ b(x)=bx mod 1\), the Hausdorff dimension of K is \(h(\tau_ b,K)/\log b\) where h is the topological entropy. Sets defined by digit restrictions correspond to “subshifts of finite type”, and it is well known that their entropies are log \(\rho\) (M). (See, for instance, R. Bowen and O.E. Lanford III [Global Analysis, Proc. Symp. Pure Math. 14, 43-49 (1970; Zbl 0211.565)].)