Expansions in non-integer bases: lower, middle and top orders

Let $q\in(1,2)$; it is known that each $x\in[0,1/(q-1)]$ has an expansion of the form $x=\sum_{n=1}^\infty a_nq^{-n}$ with $a_n\in\{0,1\}$. It was shown in \cite{EJK} that if $q<(\sqrt5+1)/2$, then each $x\in(0,1/(q-1))$ has a continuum of such expansions; however, if $q>(\sqrt5+1)/2$, then there exist infinitely many $x$ having a unique expansion \cite{GS}. In the present paper we begin the study of parameters $q$ for which there exists $x$ having a fixed finite number $m>1$ of expansions in base $q$. In particular, we show that if $q<q_2=1.71...$, then each $x$ has either 1 or infinitely many expansions, i.e., there are no such $q$ in $((\sqrt5+1)/2,q_2)$. On the other hand, for each $m>1$ there exists \ga_m>0 such that for any q\in(2-\ga_m,2), there exists $x$ which has exactly $m$ expansions in base $q$.Comment: 15 pages; to appear in J. Number Theor

Observability of rectangular membranes and plates on small sets

Since the works of Haraux and Jaffard we know that rectangular plates may be observed by subregions not satisfying the geometrical control condition. We improve these results by observing only on an arbitrarily short segment inside the domain. The estimates may be strengthened by observing on several well-chosen segments. In the second part of the paper we establish various observability theorems for rectangular membranes by applying Mehrenberger's recent generalization of Ingham's theorem.Comment: 22 pages, 8 figure

Moving and oblique observations of beams and plates

We study the observability of the one-dimensional Schr{\"o}dinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at the end of the paper