8,179 research outputs found
Digit frequencies and self-affine sets with non-empty interior
In this paper we study digit frequencies in the setting of expansions in
non-integer bases, and self-affine sets with non-empty interior.
Within expansions in non-integer bases we show that if
then every has a simply
normal -expansion. We also prove that if
then every has a
-expansion for which the digit frequency does not exist, and a
-expansion with limiting frequency of zeros , where is any real
number sufficiently close to .
For a class of planar self-affine sets we show that if the horizontal
contraction lies in a certain parameter space and the vertical contractions are
sufficiently close to then every nontrivial vertical fibre contains an
interval. Our approach lends itself to explicit calculation and give rise to
new examples of self-affine sets with non-empty interior. One particular
strength of our approach is that it allows for different rates of contraction
in the vertical direction
The growth rate and dimension theory of beta-expansions
In a recent paper of Feng and Sidorov they show that for
the set of -expansions grows
exponentially for every . In this paper we study
this growth rate further. We also consider the set of -expansions from a
dimension theory perspective
An analogue of Khintchine's theorem for self-conformal sets
Khintchine's theorem is a classical result from metric number theory which
relates the Lebesgue measure of certain limsup sets with the
convergence/divergence of naturally occurring volume sums. In this paper we ask
whether an analogous result holds for iterated function systems (IFSs). We say
that an IFS is approximation regular if we observe Khintchine type behaviour,
i.e., if the size of certain limsup sets defined using the IFS is determined by
the convergence/divergence of naturally occurring sums. We prove that an IFS is
approximation regular if it consists of conformal mappings and satisfies the
open set condition. The divergence condition we introduce incorporates the
inhomogeneity present within the IFS. We demonstrate via an example that such
an approach is essential. We also formulate an analogue of the Duffin-Schaeffer
conjecture and show that it holds for a set of full Hausdorff dimension.
Combining our results with the mass transference principle of Beresnevich and
Velani \cite{BerVel}, we prove a general result that implies the existence of
exceptional points within the attractor of our IFS. These points are
exceptional in the sense that they are "very well approximated". As a corollary
of this result, we obtain a general solution to a problem of Mahler, and prove
that there are badly approximable numbers that are very well approximated by
quadratic irrationals.
The ideas put forward in this paper are introduced in the general setting of
IFSs that may contain overlaps. We believe that by viewing IFS's from the
perspective of metric number theory, one can gain a greater insight into the
extent to which they overlap. The results of this paper should be interpreted
as a first step in this investigation
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