14 research outputs found
Sets of bounded discrepancy for multi-dimensional irrational rotation
We study bounded remainder sets with respect to an irrational rotation of the
-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who
characterized the intervals with bounded remainder in dimension one.
First we extend to several dimensions the Hecke-Ostrowski result by
constructing a class of -dimensional parallelepipeds of bounded remainder.
Then we characterize the Riemann measurable bounded remainder sets in terms of
"equidecomposability" to such a parallelepiped. By constructing invariants with
respect to this equidecomposition, we derive explicit conditions for a polytope
to be a bounded remainder set. In particular this yields a characterization of
the convex bounded remainder polygons in two dimensions. The approach is used
to obtain several other results as well.Comment: To appear in Geometric And Functional Analysi
On pair correlation and discrepancy
We say that a sequence in has Poissonian pair
correlations if
\begin{equation*}
\lim_{N \rightarrow \infty} \frac{1}{N} \# \left\{ 1 \leq l \neq m \leq N \,
: \, \left\lVert x_l-x_m \right\rVert < \frac{s}{N} \right\} = 2s
\end{equation*} for all . In this note we show that if the convergence
in the above expression is - in a certain sense - fast, then this implies a
small discrepancy for the sequence . As an easy consequence
it follows that every sequence with Poissonian pair correlations is uniformly
distributed in .Comment: To appear in Archiv der Mathemati
A positive lower bound for
Nearly 60 years ago, Erd\H{o}s and Szekeres raised the question of whether
for all irrationals . Despite its simple formulation, the question has
remained unanswered. It was shown by Lubinsky in 1999 that the answer is yes if
has unbounded continued fraction coefficients, and it was suggested
that the answer is yes in general. However, we show in this paper that for the
golden ratio ,
providing a negative answer to this long-standing open problem
F. Wiener's trick and an extremal problem for
For , let denote the classical Hardy space of the unit
disc. We consider the extremal problem of maximizing the modulus of the th
Taylor coefficient of a function which satisfies
and for some . In particular, we provide a complete
solution to this problem for and . We also study F. Wiener's
trick, which plays a crucial role in various coefficient-related extremal
problems for Hardy spaces