182 research outputs found
Forms of differing degrees over number fields
Consider a system of polynomials in many variables over the ring of integers
of a number field . We prove an asymptotic formula for the number of
integral zeros of this system in homogeneously expanding boxes. As a
consequence, any smooth and geometrically integral variety satisfies the Hasse principle, weak approximation and the
Manin-Peyre conjecture, if only its dimension is large enough compared to its
degree.
This generalizes work of Skinner, who considered the case where all
polynomials have the same degree, and recent work of Browning and Heath-Brown,
who considered the case where . Our main tool is Skinner's number
field version of the Hardy-Littlewood circle method. As a by-product, we point
out and correct an error in Skinner's treatment of the singular integral.Comment: 23 pages; minor revision; to appear in Mathematik
Asymptotic normality of additive functions on polynomial sequences in canonical number systems
The objective of this paper is the study of functions which only act on the
digits of an expansion. In particular, we are interested in the asymptotic
distribution of the values of these functions. The presented result is an
extension and generalization of a result of Bassily and K\'atai to number
systems defined in a quotient ring of the ring of polynomials over the
integers.Comment: 17 page
Construction of normal numbers via pseudo-polynomial prime sequences
In the present paper we construct normal numbers in base by concatenating
-ary expansions of pseudo polynomials evaluated at the primes. This extends
a recent result by Tichy and the author
Uniform Distribution of Prime Powers and sets of Recurrence and van der Corput sets in Z^k
We establish new results on sets of recurrence and van der Corput sets in Z^k
which refine and unify some of the previous results obtained by Sarkozy,
Furstenberg, Kamae and Mendes France, and Bergelson and Lesigne. The proofs
utilize a general equidistribution result involving prime powers which is of
independent interest
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