Berichtigung zu: Über einige Spezialisierungen der Hilbertschen Modulformen in 2 Variablen.
Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.
We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange’s result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices.
Let be a self-similar set with similarities ratio and Hausdorff dimension , let be a probability vector. The Besicovitch-type subset of is defined aswhere is the indicator function of the set . Let and be a gauge function, then we prove in this paper:(i) If , thenmoreover both of and are finite positive;(ii) If is a positive probability vector other than , then the gauge functions can be partitioned as follows
Let be a real algebraic number of degree over whose conjugates are not real. There exists an unit of the ring of integer of for which it is possible to describe the set of all best approximation vectors of .’
It is already known that all Pisot numbers are beta numbers, but for Salem numbers this was proved just for the degree 4 case. In 1945, R. Salem showed that for any Pisot number θ we can construct a sequence of Salem numbers which converge to θ. In this short note, we give some results on the beta expansion for infinitely many sequences of Salem numbers obtained by this construction.