We prove the algebraic eigenvalue conjecture of J.Dodziuk, P.Linnell, V.Mathai, T.Schick and S.Yates (see [2]) for sofic groups. Moreover, we give restrictions on the spectral measure of elements in the integral group ring. Finally, we prove a quantization of the operator norm below 2. To the knowledge of the author, there is no group known, which is not sofic.

This text is devoted to simultaneous approximation to ξ and ξ by rational numbers with the same denominator, where ξ is a non-quadratic real number. We focus on an exponent β0(ξ) that measures the quality of such approximations (when they are exceptionally good). We prove that β0(ξ) takes the same set of values as a combinatorial quantity that measures the abundance of palindrome prefixes in an...

Addition to Lemma 81. In [1], § 4, there is a variant of the matrix formula (64) for the simple continued fraction of a real number. Given integers a0, a1, . . . with ai > 0 for i ≥ 1 and writing, for n ≥ 0, as usual, pn/qn = [a0, a1, . . . , an], one checks, by induction on n, the two formulae ( 1 a0 0 1 )( 1 0 a1 1 ) · · · ( 1 an 0 1 ) = ( pn−1 pn qn−1 qn ) if n is even ( 1 a0 0 1 )( 1 0 a1 1...

The first topic of the workshop, Diophantine approximation, has at its core the study of rational numbers which closely approximate a given real number. This topic has an ancient history, going back at least to the first rational approximations for π. The adjective Diophantine comes from the third century Hellenistic mathematician Diophantus, who wrote an influential text solving various equati...

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, bo...

The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, ...

We give a theoretical description of a new homomorphic encryption scheme DA-Encrypt that is based on (non-archimedean) Diophantine Approximation.