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Let
$\mathcal {P}(\mathbf{N})$
be the power set of N. We say that a function
$\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$
is an upper density if, for all X, Y ⊆ N and h, k ∈ N+, the following hold: (f1)
$\mu ^\ast (\mathbf{N}) = 1$
; (f2)
$\mu ^\ast (X) \le \mu ^\ast (Y)$
if X ⊆ Y; (f3)
$\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$
; (f4)
$\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$
, where k · X : = {kx: x ∈ X}; and (f5)
$\mu ^\ast (X + h) = \mu ^\ast (X)$
. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)–(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that
$\mu ^\ast (X)\le 1$
for every X ⊆ N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.
We investigate the growth rate of the Birkhoff sums
$S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$
, where
$f$
is a continuous function with zero mean defined on the unit circle
$\mathbb{T}$
and
$(\unicode[STIX]{x1D6FC},x)$
is a ‘typical’ element of
$\mathbb{T}^{2}$
. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.
Relying on results due to Shmerkin and Solomyak, we show that outside a zero-dimensional set of parameters, for every planar homogeneous self-similar measure
$\unicode[STIX]{x1D708}$
, with strong separation, dense rotations and dimension greater than
$1$
, there exists
$q>1$
such that
$\{P_{z}\unicode[STIX]{x1D708}\}_{z\in S}\subset L^{q}(\mathbb{R})$
. Here
$S$
is the unit circle and
$P_{z}w=\langle z,w\rangle$
for
$w\in \mathbb{R}^{2}$
. We then study such measures. For instance, we show that
$\unicode[STIX]{x1D708}$
is dimension conserving in each direction and that the map
$z\rightarrow P_{z}\unicode[STIX]{x1D708}$
is continuous with respect to the weak topology of
$L^{q}(\mathbb{R})$
.
By using methods of subordinacy theory, we study packing continuity properties of spectral measures of discrete one-dimensional Schrödinger operators acting on the whole line. Then we apply these methods to Sturmian operators with rotation numbers of quasibounded density to show that they have purely
$\unicode[STIX]{x1D6FC}$
-packing continuous spectrum. A dimensional stability result is also mentioned.
Suppose that
$0<|\unicode[STIX]{x1D70C}|<1$
and
$m\geqslant 2$
is an integer. Let
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$
be the self-similar measure defined by
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$
. Assume that
$\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$
for some
$p,q,r\in \mathbb{N}^{+}$
with
$(p,q)=1$
and
$(p,m)=1$
. We prove that if
$(q,m)=1$
, then there are at most
$m$
mutually orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
and
$m$
is the best possible. If
$(q,m)>1$
, then there are any number of orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
.
We construct a family of self-affine tiles in
$\mathbb{R}^{d}$
(
$d\geqslant 2$
) with noncollinear digit sets, which naturally generalizes a class studied originally by Q.-R. Deng and K.-S. Lau in
$\mathbb{R}^{2}$
, and its extension to
$\mathbb{R}^{3}$
by the authors. We obtain necessary and sufficient conditions for the tiles to be connected and for their interiors to be contractible.
We describe how to approximate fractal transformations generated by a one-parameter family of dynamical systems
$W:[0,1]\rightarrow [0,1]$
constructed from a pair of monotone increasing diffeomorphisms
$W_{i}$
such that
$W_{i}^{-1}:[0,1]\rightarrow [0,1]$
for
$i=0,1$
. An algorithm is provided for determining the unique parameter value such that the closure of the symbolic attractor
$\overline{\unicode[STIX]{x1D6FA}}$
is symmetrical. Several examples are given, one in which the
$W_{i}$
are affine and two in which the
$W_{i}$
are nonlinear. Applications to digital imaging are also discussed.
This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous
$f$
-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the
$f$
-divergence is proposed, and related functional affine isoperimetric inequalities are established.
We consider a one-parameter family of dynamical systems
$W:[0,1]\rightarrow [0,1]$
constructed from a pair of monotone increasing diffeomorphisms
$W_{i}$
such that
$W_{i}^{-1}:$
$[0,1]\rightarrow [0,1]$
$(i=0,1)$
. We characterise the set of symbolic itineraries of
$W$
using an attractor
$\overline{\unicode[STIX]{x1D6FA}}$
of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which
$\overline{\unicode[STIX]{x1D6FA}}$
is symmetrical.
Let
$\mathbf{M}=(M_{1},\ldots ,M_{k})$
be a tuple of real
$d\times d$
matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether
$\mathbf{M}$
possesses the following property: there exist two constants
$\unicode[STIX]{x1D706}\in \mathbb{R}$
and
$C>0$
such that for any
$n\in \mathbb{N}$
and any
$i_{1},\ldots ,i_{n}\in \{1,\ldots ,k\}$
, either
$M_{i_{1}}\cdots M_{i_{n}}=\mathbf{0}$
or
$C^{-1}e^{\unicode[STIX]{x1D706}n}\leq \Vert M_{i_{1}}\cdots M_{i_{n}}\Vert \leq Ce^{\unicode[STIX]{x1D706}n}$
, where
$\Vert \cdot \Vert$
is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on
$\mathbb{R}$
, the absolute continuity of certain self-affine measures in
$\mathbb{R}^{d}$
and the dimensional regularity of a class of sofic affine-invariant sets in the plane.
For
$x\in (0,1]$
and a positive integer
$n,$
let
$S_{\!n}(x)$
denote the summation of the first
$n$
digits in the dyadic expansion of
$x$
and let
$r_{n}(x)$
denote the run-length function. In this paper, we obtain the Hausdorff dimensions of the following sets:
where
$\unicode[STIX]{x1D6FE}$
ranges over all closed geodesics
$\unicode[STIX]{x1D6FE}:\mathbb{S}^{1}\rightarrow \mathbb{T}^{2}$
and
$|\unicode[STIX]{x1D6FE}|$
denotes its length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic
$\unicode[STIX]{x1D6FE}$
attaining the supremum in terms of the smoothness of the function: for all
$s\geq 2$
,
We exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.
The class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.
where
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$
,
$u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$
and
$s>n/2$
. The inequality fails for
$s=n/2$
. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if
$n\geqslant 3$
and
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$
is bounded, then
where
$L^{p,q}$
is the Lorentz space refinement of
$L^{p}$
. This inequality fails for
$n=2$
, and we prove a sharp substitute result: there exists
$c>0$
such that for all
$\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$
with finite measure,
This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.
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
$\unicode[STIX]{x1D6FD}\in (1,1.787\ldots )$
then every
$x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$
has a simply normal
$\unicode[STIX]{x1D6FD}$
-expansion. We also prove that if
$\unicode[STIX]{x1D6FD}\in (1,(1+\sqrt{5})/2)$
then every
$x\in (0,1/(\unicode[STIX]{x1D6FD}-1))$
has a
$\unicode[STIX]{x1D6FD}$
-expansion for which the digit frequency does not exist, and a
$\unicode[STIX]{x1D6FD}$
-expansion with limiting frequency of zeros
$p$
, where
$p$
is any real number sufficiently close to
$1/2$
. 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
$1$
, then every non-trivial vertical fibre contains an interval. Our approach lends itself to explicit calculation and gives 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.
In this paper we discuss some dimension results for triangle sets of compact sets in
$\mathbb{R}^{2}$
. In particular we prove that for any compact set
$F$
in
$\mathbb{R}^{2}$
, the triangle set
$\unicode[STIX]{x1D6E5}(F)$
satisfies
We establish several new metrical results on the distribution properties of the sequence ({xn})n≥1, where {·} denotes the fractional part. Many of them are presented in a more general framework, in which the sequence of functions (x ↦ xn)n≥1 is replaced by a sequence (fn)n≥1, under some growth and regularity conditions on the functions fn.
We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case (
$G(x,p)=\int c(x,y)dp$
) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.