62 research outputs found
A variant of the Hales-Jewett Theorem
It was shown by V. Bergelson that any set B with positive upper
multiplicative density contains nicely intertwined arithmetic and geometric
progressions: For each positive integer k there exist integers a,b,d such that
{b(a+id)^j:i,j \in\nhat k}\subset B. In particular one cell of each finite
partition of the positive integers contains such configurations. We prove a
Hales-Jewett type extension of this partition theorem
Strong characterizing sequences of countable groups
Andr\'as Bir\'o and Vera S\'os prove that for any subgroup of \T
generated freely by finitely many generators there is a sequence
such that for all \beta \in \T we have ( denotes the distance to the
nearest integer) We extend this result to arbitrary countable
subgroups of \T. We also show that not only the sum of norms but the sum of
arbitrary small powers of these norms can be kept small. Our proof combines
ideas from the above article with new methods, involving a filter
characterization of subgroups of \T
Martingale Inequalities and Deterministic Counterparts
We study martingale inequalities from an analytic point of view and show that
a general martingale inequality can be reduced to a pair of deterministic
inequalities in a small number of variables. More precisely, the optimal bound
in the martingale inequality is determined by a fixed point of a simple
nonlinear operator involving a concave envelope. Our results yield an
explanation for certain inequalities that arise in mathematical finance in the
context of robust hedging.Comment: 22 page
Sequences and filters of characters characterizing subgroups of compact abelian groups
Let H be a countable subgroup of the metrizable compact abelian group G and
f:H -> T=R/Z a (not necessarily continuous) character of H. Then there exists a
sequence (chi_n)_n of (continuous) characters of G such that lim_n chi_n(alpha)
= f(alpha) for all alpha in H and (chi_n(alpha))_n does not converge whenever
alpha in G\H. If one drops the countability and metrizability requirement one
can obtain similar results by using filters of characters instead of sequences.
Furthermore the introduced methods allow to answer questions of Dikranjan et
al.Comment: laTex2e, 10 page
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