Univoque bases of real numbers: Simply normal bases, irregular bases and multiple rationals

Given a positive integer M and real number x∈(0,1], we call q∈(1,M+1] univoque simply normal base of x if there exists unique sequence (di)∈{0,1,…,M}N such that x=∑i=1∞diq−i. Similarly, is called irregular x=∑i=1∞diq−i the (di) has no digit frequency. Let USN(x) UIr(x) be sets bases x, respectively. In this paper show for any x∈(0,1] both have full Hausdorff dimension. Furthermore, given finitely many rationals x1,x2,…,xℓ∈(0,1] so each xi finite expansion in M+1, dimensional set q.