The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated
by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers
1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions
were characterized in  and this characterization led to the determination (in ) of the smallest number q having this
curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that
this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer
bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all
of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section
we answer a question concerning the existence of universal expansions, a notion introduced in .
Authors:Claudio Baiocchi, Vilmos Komornik, and Paola Loreti
Completing a series of works begun by Wiener , Paley and Wiener  and Ingham , a far-reaching generalization of
Parseval"s identity was obtained by Beurling  for nonharmonic Fourier series whose exponents satisfy a uniform gap condition.
Later this gap condition was weakened by Ullrich , Castro and Zuazua , Jaffard, Tucsnak and Zuazua  and then in
 in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying
a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained
in  and . Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system
of vibrating strings.