Expansions in non-integer base
Anna Chiara Lai · OpenGrey (Institut de l'Information Scientifique et Technique) · 2010
This thesis is devoted to the study of developments in series of powers in real or complex base and with coefficients belonging to a fmite set of non-negative real numbers, named alphabet. When a number x is the sum of a development in series of powers with a base q and alphabet, we say that x is representable in base q and alphabet A. We study the representability in complex base by studying the convex hull of the set of the representable numbers when the base has a rational argument. As the convexity of the set of representable numbers is a sufficient condition for the full representability, such a result gives a new class of numeration Systems in complex base. We then study the numeration System in negative base —q and alphabet A={0,1,. . . ,[q]} and the class of -q-expansions. In particular we extend to the negative case some well known results on the computation of the entropy, the recognisability by fmite automata and the existence of fmite automata performing some arithmetic operations. Finally we assume the base to be real and positive and we study the redundancy of the representations with arbitrary alphabets. We prove the existence of a sort of generalized Golden Mean for arbitrary alphabets, namely we show that expansions are never unique if and only if the base is chosen below a critical value. In the case of a ternary alphabet we explicitely characterize such a critical base as well as the unique expansions for sufficiently small bases.