Unique double base expansions

Abstract

For two real bases \(q_0, q_1 > 1\) , we consider expansions of real numbers of the form \(\sum _{k=1}^{\infty } i_k/(q_{i_1}q_{i_2}\ldots q_{i_k})\) with \(i_k \in \{0,1\}\) , which we call \((q_0,q_1)\) -expansions. A sequence \((i_k)\) is called a unique \((q_0,q_1)\) -expansion if all other sequences have different values as \((q_0,q_1)\) -expansions, and the set of unique \((q_0,q_1)\) -expansions is denoted by \(U_{q_0,q_1}\) . In the special case \(q_0 = q_1 = q\) , the set \(U_{q,q}\) is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases \((q_0, q_1)\) with trivial \(U_{q_0,q_1}\) from those with non-trivial \(U_{q_0,q_1}\) is the graph of a function \(\mathcal {G}(q_0)\) that we call generalized golden ratio. Similarly, the curve separating pairs \((q_0, q_1)\) with countable \(U_{q_0,q_1}\) from those with uncountable \(U_{q_0,q_1}\) is the graph of a function \(\mathcal {K}(q_0)\) that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in \(q_0\) and \(q_1\) , that \(\mathcal {G}\) and \(\mathcal {K}\) are continuous, strictly decreasing, hence almost everywhere differentiable on \((1,\infty )\) , and that the Hausdorff dimension of the set of \(q_0\) satisfying \(\mathcal {G}(q_0)=\mathcal {K}(q_0)\) is zero. We give formulas for \(\mathcal {G}(q_0)\) and \(\mathcal {K}(q_0)\) for all \(q_0 > 1\) , using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems.