Unique expansion of points of a class of self-similar sets with overlaps (Q2902665)

Let \(q>1\) and consider the iterated function system (IFS) defined by \(\phi_k(x):= (x+k)/q\) for \(k\in \{0,1,q\}\). It is well known that points \(x\) in the self similar set \(F_q\) generated by the IFS\(\{\phi_0,\phi_1,\phi_q\}\) (recall that \(F_q\) is the unique non-empty compact set invariant under the IFS\(\{\phi_0,\phi_1,\phi_q\}\)) can be encoded by digits from \(\{0,1,q\}\), and that some \(x \in F_q\) may have multiple (even infinitely many) \(q\)-expansions. This IFS fails to satisfy the open set condition, so \(F_q\) is a self similar set with overlaps.NEWLINENEWLINEThe authors compute the exact Hausdorff dimension of \(\mathcal U_q:=\{x\in F_q: x\) has a unique \(q\)-expansion for all \(q \geq (3+\sqrt 5)/2\) (see Theorem 1.1 (c)). Their result is similar to that of \textit{P. Glendinning} and \textit{N. Sidorov} [Math. Res. Lett. 8, 535--543 (2001; Zbl 1002.11009)], but more difficult and requiring new techniques. While the Hausdorff dimension of \(F_q\) is known for all \(q>1\) by \textit{S.-M. Ngai} and \textit{Y. Wang} [J. Lond. Math. Soc. 63, 655--672 (2001; Zbl 1013.28008)], the Hausdorff dimension of \(\mathcal U_q\) for \(q_c<q<(3+\sqrt 5)/2\) is still unknown, where \(q_c\) is the positive solution of the equation \(x^3-3x^2+2x-1=0\). However, for such \(q\) the authors prove that \(\mathcal U_q\) is uncountable (Theorem 1.1 (b)); moreover, for \(1<q \leq q_c\) the set \(\mathcal U_q\) shrinks to \(\{0,q/(q-1)\}\) (see Theorem 1.1 (a)).