Positive Hausdorff Dimension Research Articles

A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties).

PARTITION GENERICITY AND PIGEONHOLE BASIS THEOREMS

AbstractThere exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions of weakness. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems.

Nonuniqueness and Existence of Continuous, Globally Dissipative Euler Flows

We show that Hölder continuous incompressible Euler flows that satisfy the local energy inequality (“globally dissipative” solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is $$C^0$$ dense in the space of continuous, divergence free vector fields on the torus $${{\mathbb {T}}}^3$$ . The construction of these solutions involves a new and explicit convex integration approach mirroring Kraichnan’s LDIA theory of turbulent energy cascades that overcomes the limitations of previous schemes, which had been restricted to bounded measurable solutions or to continuous solutions that dissipate total kinetic energy.

Sobolev norms explosion for the cubic NLS on irrational tori

We consider the cubic nonlinear Schrödinger equation on 2-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every s>0, s≠1 and almost every choice of spatial periods we construct solutions whose Hs Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion.Note that the irrationality of the space periods decouples the linear resonant interactions into products of 1-dimensional resonances, reducing considerably the complexity of the resonant dynamics usually used to construct transfer of energy solutions.However, one can provide these growth of Sobolev norms solutions by using quasi-resonances relying on Diophantine approximation properties of the space periods.

Intersections of thick compact sets in $${\mathbb {R}}^d$$

AbstractWe introduce a definition of thickness in $${\mathbb {R}}^d$$ R d and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt’s game. As an application we prove that given any compact set in $${\mathbb {R}}^d$$ R d with thickness $$\tau $$ τ , there is a number $$N(\tau )$$ N ( τ ) such that the set contains a translate of all sufficiently small similar copies of every set in $${\mathbb {R}}^d$$ R d with at most $$N(\tau )$$ N ( τ ) elements; indeed the set of such translations has positive Hausdorff dimension. We also prove a gap lemma and bounds relating Hausdorff dimension and thickness.

Lebesgue measure of Feigenbaum Julia sets

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ the corresponding set of parameters $c$ is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.

Patterns in sets of positive density in trees and affine buildings

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.

On the smallest base in which a number has a unique expansion

Given a real number x > 0 x>0 , we determine q s ( x ) ≔ inf ⁡ U ( x ) q_s(x)≔\operatorname {inf}{\mathscr {U}}(x) , where U ( x ) {\mathscr {U}}(x) is the set of all bases q ∈ ( 1 , 2 ] q\in (1,2] for which x x has a unique expansion of 0 0 ’s and 1 1 ’s. We give an explicit description of q s ( x ) q_s(x) for several regions of x x -values. For others, we present an efficient algorithm to determine q s ( x ) q_s(x) and the lexicographically smallest unique expansion of x x . We show that the infimum is attained for almost all x x , but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function q s q_s is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of q s q_s . A large part of the paper is devoted to the level sets L ( q ) ≔ { x > 0 : q s ( x ) = q } L(q)≔\{x>0:q_s(x)=q\} . We show that L ( q ) L(q) is finite for almost every q q , but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant q K L = min ⁡ U ( 1 ) ≈ 1.787 q_{KL}=\operatorname {min}{\mathscr {U}}(1)\approx 1.787 we prove that L ( q K L ) L(q_{KL}) has both infinitely many left- and infinitely many right accumulation points.

Small values of Weyl sums

We prove that the set of (x1,…,xd)∈[0,1)d, such thatlim_N→∞|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|=0, contains a dense Gδ set in [0,1)d and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums∑n=1Nexp⁡(2πixnd),x∈[0,1).

Uncertainty Principles for Fourier Multipliers

The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal $$u=1/w$$ is a (p, q)-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift-invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian–Low uncertainty principle in these settings.

Univoque bases of real numbers: Local dimension, Devil's staircase and isolated points

Given a positive integer M and a real number x>0, let U(x) be the set of all bases q∈(1,M+1] for which there exists a unique sequence (di)=d1d2… with each digit di∈{0,1,…,M} satisfyingx=∑i=1∞diqi. The sequence (di) is called a q-expansion of x. In this paper we investigate the local dimension of U(x) and prove a ‘variation principle’ for unique non-integer base expansions. We also determine the critical values of U(x) such that when x passes the first critical value the set U(x) changes from a set with positive Hausdorff dimension to a countable set, and when x passes the second critical value the set U(x) changes from an infinite set to a singleton. Denote by U(x) the set of all unique q-expansions of x for q∈U(x). We give the Hausdorff dimension of U(x) and show that the dimensional function x↦dimH⁡U(x) is a non-increasing Devil's staircase. Finally, we investigate the topological structure of U(x). Although the set U(1) has no isolated points, we prove that for typical x>0 the set U(x) contains isolated points.

Critical base for the unique codings of fat Sierpinski gasket

Given β ∈ (1, 2) the fat Sierpinski gasket is the self-similar set in generated by the iterated function system (IFS) Then for each point there exists a sequence such that , and the infinite sequence (di) is called a coding of P. In general, a point in may have multiple codings since the overlap region has non-empty interior, where Δβ is the convex hull of . In this paper we are interested in the invariant set Then each point in has a unique coding. We show that there is a transcendental number βc ≈ 1.552 63 related to the Thue–Morse sequence, such that has positive Hausdorff dimension if and only if β > βc. Furthermore, for β = βc the set is uncountable but has zero Hausdorff dimension, and for β < βc the set is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of .

Restriction of Laplace–Beltrami Eigenfunctions to Arbitrary Sets on Manifolds

Abstract Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace–Beltrami eigenfunctions when restricted to a wide variety of subsets $\Gamma $ of $M$. The sets $\Gamma $ that we consider are Borel measurable, Lebesguenull but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $\Gamma $. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in [ 8, 18, 19, 32]. Under an additional measure-theoretic assumption on $\Gamma $, the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.

On large values of Weyl sums

A special case of the Menshov–Rademacher theorem implies for almost all polynomials x1Z+…+xdZd∈R[Z] of degree d for the Weyl sums satisfy the upper bound|∑n=1Nexp⁡(2πi(x1n+…+xdnd))|⩽N1/2+o(1),N→∞. Here we investigate the exceptional sets of coefficients (x1,…,xd) with large values of Weyl sums for infinitely many N, and show that in terms of the Baire categories and Hausdorff dimension they are quite massive, in particular of positive Hausdorff dimension in any fixed cube inside of [0,1]d. We also use a different technique to give similar results for sums with just one monomial xnd. We apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well.

Pro‐p groups of positive rank gradient and Hausdorff dimension

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$ that are subnormal in~$G$ via finitely generated successive quotients. This pro-$p$ version of a well-known theorem of Greenberg generalises similar assertions that were known to hold for non-abelian free pro-$p$ groups, non-soluble Demushkin pro-$p$ groups and other related pro-$p$ groups. The result we prove is reminiscent of Gaboriau's theorem for countable groups with positive first $\ell^2$-Betti number, but not quite a direct analogue. The approach via the notion of Hausdorff dimension in pro-$p$ groups also leads to our main results. We show that every finitely generated pro-$p$ group $G$ of positive rank gradient has full Hausdorff spectrum $\text{hspec}^\mathcal{F}(G) = [0,1]$ with respect to the Frattini series~$\mathcal{F}$. Using different, Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups $G$ have full Hausdorff spectrum $\text{hspec}^\mathcal{Z}(G) = [0,1]$ with respect to the Zassenhaus series~$\mathcal{Z}$. This resolves a long-standing problem in the subject of Hausdorff dimensions in pro-$p$ groups. The results about full Hausdorff spectra hold more generally for finite direct products of finitely generated pro-$p$ groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups, respectively. Analogously, the aforementioned results with respect to the Frattini series generalise further to Hausdorff dimension functions with respect to arbitrary iterated verbal filtrations. Finally, we determine the normal Hausdorff spectra of such direct products.

Extracting randomness within a subset is hard

The tree forcing method of Liu enables the cone avoiding of bounded enumeration of a given tree, within subsets or co-subsets of an arbitrary given set, provided the given tree does not admit computable bounded enumeration. Using this result, he settled and reproduced a series of problems and results in reverse mathematics and the theory of algorithmic randomness, including showing that every 1-random set has an infinite subset or co-subset which computes no 1-random set. In this paper, we show that for any given 1-random set A, there exists an infinite subset G of A such that G does not compute any set with positive effective Hausdorff dimension. In particular, we answer in the affirmative Kjos-Hanssen’s 2006 question whether each 1-random set has an infinite subset which computes no 1-random set. The result is surprising in that the tree forcing technique seems to heavily rely on subset co-subset combinatorics, whereas this result does not.

Positive Hausdorff Dimensional Spectrum for the Critical Almost Mathieu Operator

We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.

Uniformly de Bruijn Sequences and Symbolic Diophantine Approximation on Fractals

Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in {k geq 2} letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given k, these sequences bear a strong connection to Diophantine approximation on certain fractals. In particular, the optimality of an intrinsic Dirichlet function on these fractals with respect to the height function defined by symbolic representations of rationals follows from these results.

Localisation of Bochner Riesz means on sets of positive Hausdorff dimension in \mathbb{R}^d

Localisation of Bochner Riesz means on sets of positive Hausdorff dimension in \mathbb{R}^d

On univoque and strongly univoque sets

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet {0,1,…,α} and a real number (base) 1<β<α+1, the so-called univoque set of numbers which have a unique expansion in base β has garnered a great deal of attention in recent years. Motivated by recent applications of β-expansions to Bernoulli convolutions and a certain class of self-affine functions, we introduce the notion of a strongly univoque set. We study in detail the set Dβ of numbers which are univoque but not strongly univoque. Our main result is that Dβ is nonempty if and only if the number 1 has a unique nonterminating expansion in base β, and in that case, Dβ is uncountable. We give a sufficient condition for Dβ to have positive Hausdorff dimension, and show that, on the other hand, there are infinitely many values of β for which Dβ is uncountable but of Hausdorff dimension zero.