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In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form ∑ i h - 1 (q h (μ (B (x i , r i))) + t g (r i)) , where μ represents a Borel probability measure on R d , and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework. [ABSTRACT FROM AUTHOR]

A Trott number is a number x ∈ (0 , 1) whose continued fraction expansion is equal to its base b expansion for a given base b, in the following sense: If x = [ 0 ; a 1 , a 2 , ⋯ ] , then x = (0. a ^ 1 a ^ 2 ⋯) b , where a ^ i is the string of digits resulting from writing a i in base b. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set T b of Trott numbers is a complete G δ set. We prove moreover that the union T : = ⋃ b ≥ 2 T b is nowhere dense and has Hausdorff dimension less than one. Finally, we give several sufficient conditions on bases b and b ′ such that T b ∩ T b ′ = ∅ , and conjecture that this is the case for all b ≠ b ′ . This question has connections with some deep theorems in Diophantine approximation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that if the Hewitt–Stromberg pre-measures with respect to the gauge are finite, then these pre-measures have a kind of outer regularity in a general metric space X. We give also some conditions on the Hewitt–Stromberg pre-measures with respect to the gauge such that the Hewitt–Stromberg measures have an almost inner regularity on a complete separable metric space X. [ABSTRACT FROM AUTHOR]

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples. [ABSTRACT FROM AUTHOR]

The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet's theorem, the other follows from Minkowski's theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work. [ABSTRACT FROM AUTHOR]

A Radon measure μ is n-rectifiable if it is absolutely continuous with respect to H n and μ -almost all of supp μ can be covered by Lipschitz images of R n . In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called α and β 2 numbers. The second one involves α 2 numbers—coefficients quantifying flatness via Wasserstein distance W 2 . Both conditions are necessary for rectifiability, too—the first one was shown to be necessary by Tolsa, while the necessity of the α 2 condition is established in our recent paper. Thus, we get two new characterizations of rectifiability. [ABSTRACT FROM AUTHOR]

In this paper, we present the duality theory for general weighted spaces of vector functions C V 0 (X , E) where X is a Hausdorff locally compact space and E is a Hausdorff locally convex space. We mention that a characterization of the dual of C V 0 (X , E) in the particular case V ⊂ C + (X , R) (i.e., for any weight v ∈ V there is a continuous function w ∈ C + (X , R) such that v ≤ w ), is mentioned by Prolla (Math. Anal. 191:283–289, 1971). Also we extend de Branges Lemma in this new setting for vector subspaces of a weighted space of vector functions. Using this theorem, we find various approximations results for weighted spaces of vector functions.In the particular case when E is a locally convex lattice, then the weighted space of vector function C V 0 (X , E) is also a locally convex lattice. We present in this case a characterization theorem of closed ordered ideals in C V 0 (X , E). [ABSTRACT FROM AUTHOR]

In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley et al. (Math Ann 338(1):97–118, 2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg's times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured. [ABSTRACT FROM AUTHOR]

Working with sets of infinite Ψ t -Hausdorff measures can be cumbersome, so simplifying them to sets of positive finite general fractal measures proves to be highly beneficial. This paper aims to demonstrate that the Ψ t -Hausdorff measures adhere to the property of being a subset of positive and finite measures. Our main result is then applied to establish that the general fractal function dimension is defined as the supremum of the general fractal dimension across all Borel probability measures. [ABSTRACT FROM AUTHOR]

Since the publication of our paper "A note on fractal measures and Cartesian products" in this journal, we have become aware of some work that is closely related to ours. We wish to call the readers attention to this material. We thank the Editor in chief for informing us of this related work. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a Marstrand-type theorem to give the almost sure Hewitt–Stromberg and certain modified intermediate dimensions of projections of sets. We give alternative definitions of dimension profiles in terms of capacities of a set E with respect to certain kernels, which lead to the Hewitt–Stromberg and modified intermediate dimensions of projections fairly easily. [ABSTRACT FROM AUTHOR]

Given a non-increasing function ψ , let Exact (ψ) be the set of complex numbers which are approximable by complex rational numbers to order ψ but to no better order. In this paper, we obtain the Hausdorff dimension and the packing dimension of Exact (ψ) when ψ (x) = o (x - 2) . Moreover, without the condition ψ (x) = o (x - 2) , we also prove that the Hausdorff dimension of Exact (ψ) is greater than 2 - τ / (1 - 2 τ) when 0 < τ = lim sup x → + ∞ x 2 ψ (x) is small enough. [ABSTRACT FROM AUTHOR]

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators L 1 and L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of L 1 is equivalent to the same property for L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which L 2 is either the transpose of L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups H n . Among other results we prove that measures with (2 n + 1) -density have only flat tangents and conclude the classification of uniform measures in H 1 . [ABSTRACT FROM AUTHOR]

Hölder regularity, isotropic wavelet bases and Hausdorff dimension are not optimal for analyzing directional or anisotropic features for signals on R d with d ≥ 2 . This paper aims to overcome this drawback using rectangular pointwise regularity, hyperbolic wavelets and dimension prints. Rectangular pointwise regularity L (α 1 , … , α d) (x 1 , … , x d) is defined through local oscillations over rectangular parallelepipeds. It extends the alternative definition of Hölder regularity by means of isotropic local oscillations done by Jaffard. It also substitutes the rectangular pointwise Lipschitz regularity introduced by Ayache et al. and Kamont for (α 1 , … , α d) ∈ (0 , 1) d in order to cover any (α 1 , … , α d) ∈ (0 , ∞) d . It is also a pointwise version of the Kamont’s anisotropic Besov spaces on the unit cube. We prove characterizations in terms of hyperbolic wavelets. We deduce a new information on the multifractal analysis of rectangular pointwise behaviors, relating the dimension prints of sets of level rectangular pointwise regularities to average quantities expressed in terms of hyperbolic wavelet leaders. We apply our results for some anisotropic selfsimilar cascade wavelet series. [ABSTRACT FROM AUTHOR]

In this paper, we give a new product formula : H t (E) H s (F) ≤ c 1 H t + s (E × F) ≤ c 2 H s (E) P t (F) ≤ c 3 P t + s (E × F) ≤ c 4 P s (E) P t (F). where E ⊆ R d , F ⊆ R l , t , s ≥ 0 and H t and P s denote, respectively, the lower and upper Hewitt–Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt–Stromberg dimensions b (E × F) and B (E × F) in terms of the Hewitt–Strombeg dimensions of E and F. [ABSTRACT FROM AUTHOR]

Recently, we have proved that the rectangular pointwise Lipschitz regularity of a continuous function on the unit square is directly related with the local suprema of the coefficients of the function in the tensor product Faber–Schauder basis. In this paper, we provide print dimension information on the distribution at all bi-scales of these local suprema. We apply our results for self-affine functions associated to the Schauder product function and a particular type of Sierpinski carpets. [ABSTRACT FROM AUTHOR]

In this paper we show that if $$\mu $$ is a Borel measure in $${{\mathbb {R}}}^{n+1}$$ with growth of order n, such that the n-dimensional Riesz transform $${{\mathcal {R}}}_\mu $$ is bounded in $$L^2(\mu )$$ , and $$B\subset {{\mathbb {R}}}^{n+1}$$ is a ball with $$\mu (B)\approx r(B)^n$$ such that: then there exists a uniformly n-rectifiable set $$\Gamma $$ , with $$\mu (\Gamma \cap B)\gtrsim \mu (B)$$ , and such that $$\mu |_\Gamma $$ is absolutely continuous with respect to $${{\mathcal {H}}}^n|_\Gamma $$ . This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg. [ABSTRACT FROM AUTHOR]

In this paper, we shall be concerned with evaluation of Hewitt–Stromberg measure of Cartesian product sets by means of the measure of their components. This is done by the construction of new multifractal measures, on the Euclidean space R n , in a similar manner to Hewitt–Stromberg measures but using the class of all n-dimensional half-open binary cubes of covering sets in the definition rather than the class of all balls. [ABSTRACT FROM AUTHOR]

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to $${\mathbb {R}}^n$$ . The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by $$C(K,{\mathbb {R}}^n)$$ the set of continuous maps from K to $${\mathbb {R}}^n$$ endowed with the maximum norm. Let $$\dim _{*}$$ be one of the topological dimension $$\dim _T$$ , the Hausdorff dimension $$\dim _H$$ , or the packing dimension $$\dim _P$$ . Define We prove that $$d^n_{*}(K)$$ is the right notion to describe the dimensions of the fibers of a generic continuous map $$f\in C(K,{\mathbb {R}}^n)$$ . In particular, we show that $$\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)$$ provided that $$\dim _T K\ge n$$ , otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume $$\dim _T K\ge n$$ . If K is sufficiently homogeneous, then we can say much more. For example, we prove that $$\dim _{*}f^{-1}(y)=d^n_{*}(K)$$ for a generic $$f\in C(K,{\mathbb {R}}^n)$$ for all $$y\in {{\mathrm{int}}}f(K)$$ if and only if $$d^n_{*}(U)=d^n_{*}(K)$$ or $$\dim _T U

We show that there are functions f in the Hölder class $${C^{\alpha}[0,1]}$$ , $${1 < \alpha < 2}$$ such that $${f |_{A}}$$ is neither convex nor concave for any $${A\subset [0,1]}$$ with $${\overline{{\rm dim}}_\textsc{M}\,A > \alpha-1}$$ . Our earlier result shows that for the typical/generic $${f\in C_1^{\alpha}[0,1]}$$ , $${0\leq \alpha < 2}$$ there is always a set $${A\subset [0,1]}$$ such that $${f |_A}$$ is convex and $${\overline{{\rm dim}}_\textsc{M}\,A=1}$$ . The analogous statement for monotone restrictions is the following: there are functions $${f}$$ in the Hölder class $${C^{\alpha}[0,1]}$$ , $${1/2 \leq \alpha < 1}$$ such that $${f |_{A}}$$ is not monotone on $${A\subset [0,1]}$$ with $${\overline{{\rm dim}}_\textsc{M}\,A > \alpha}$$ . This statement is not true for the range of parameters $${\alpha < 1/2}$$ and the main theorem of this paper for the parameter range $${1< \alpha < 3/2}$$ cannot be obtained by integration of the result about monotone restrictions. [ABSTRACT FROM AUTHOR]

We say that a set in a Euclidean space does not contain an angle α if the angle determined by any three points of the set is not equal to α. The goal of this paper is to construct compact sets of large Hausdorff dimension that do not contain a given angle α ∈ (0, π). We will construct such sets in ℝ of Hausdorff dimension c( α) n with a positive c( α) depending only on α provided that α is different from π/3, π/2 and 2 π/3. This improves on an earlier construction (due to several authors) that has dimension c( α) log n. The main result of the paper concerns the case of the angles π/3 and 2 π/3. We present self-similar sets in ℝ of Hausdorff dimension $c{{\sqrt[3]{n}} \mathord{\left/ {\vphantom {{\sqrt[3]{n}} {\log n}}} \right. \kern-\nulldelimiterspace} {\log n}}$ with the property that they do not contain the angles π/3 and 2 π/3. The constructed sets avoid not only the given angle α but also a small neighbourhood of α. [ABSTRACT FROM AUTHOR]

For two real bases q 0 , q 1 > 1 , we consider expansions of real numbers of the form ∑ k = 1 ∞ i k / (q i 1 q i 2 ... q i k) with i k ∈ { 0 , 1 } , which we call (q 0 , q 1) -expansions. A sequence (i k) is called a unique (q 0 , q 1) -expansion if all other sequences have different values as (q 0 , q 1) -expansions, and the set of unique (q 0 , q 1) -expansions is denoted by U q 0 , q 1 . In the special case q 0 = q 1 = q , the set U q , q is trivial if q is below the golden ratio and uncountable if q is above the Komornik–Loreti constant. The curve separating pairs of bases (q 0 , q 1) with trivial U q 0 , q 1 from those with non-trivial U q 0 , q 1 is the graph of a function G (q 0) that we call generalized golden ratio. Similarly, the curve separating pairs (q 0 , q 1) with countable U q 0 , q 1 from those with uncountable U q 0 , q 1 is the graph of a function K (q 0) that we call generalized Komornik–Loreti constant. We show that the two curves are symmetric in q 0 and q 1 , that G and K are continuous, strictly decreasing, hence almost everywhere differentiable on (1 , ∞) , and that the Hausdorff dimension of the set of q 0 satisfying G (q 0) = K (q 0) is zero. We give formulas for G (q 0) and K (q 0) for all q 0 > 1 , using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of S-adic sequences including Sturmian and the Thue–Morse sequences are simpler than those of Labarca and Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems. [ABSTRACT FROM AUTHOR]

We prove a decomposition theorem of Besicovitch's type for the relative multifractal Hausdorff measure and packing measure in a probability space. By obtaining a new necessary condition for the strong regularity with the multifractal measures in a more general framework, we extend in this paper the density theorem of Dai and Li (A multifractal formalism in a probability space. Chaos Solitons Fractals 27:57–73, 2006). In particular, this result is more refined than those found in Dai and Taylor (Defining fractal in a probability space. Ill J Math 38:480–500, 1994). [ABSTRACT FROM AUTHOR]

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n − 1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one. [ABSTRACT FROM AUTHOR]

Let X be a bounded subset of Rd and write Cu(X) for the set of uniformly continuous functions on X equipped with the uniform norm. The lower and upper box dimensions, denoted by dim̲B(graph(f)) and dim¯B(graph(f)), of the graph graph(f) of a function f∈Cu(X) are defined by dim̲B(graph(f))=limδ↘0inflogNδ(graph(f))-logδ,dim¯B(graph(f))=limδ↘0suplogNδ(graph(f))-logδ,where Nδ(graph(f)) denotes the number of δ-mesh cubes that intersect graph(f).Hyde et al. have recently proved that the box counting function (∗)logNδ(graph(f))-logδof the graph of a typical function f∈Cu(X) diverges in the worst possible way as δ↘0. More precisely, Hyde et al. showed that for a typical function f∈Cu(X), the lower box dimension of the graph of f is as small as possible and if X has only finitely many isolated points, then the upper box dimension of the graph of f is as big as possible.In this paper we will prove that the box counting function (*) of the graph of a typical function f∈Cu(X) is spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in (*) divergent as δ↘0, but it is so irregular that it remains spectacularly divergent as δ↘0 even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz-Hardy logarithmic averages. For example, if the box dimension of X exists, then we show that for a typical function f∈Cu(X), all the higher order lower Hölder and Cesàro averages of the box counting function (*) are as small as possible, namely, equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1. [ABSTRACT FROM AUTHOR]

Suppose that $${{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}$$ and x = E. E E · · · is the P-Cantor series expansion of $${x \in \mathbb{R}}$$ . We define The functions $${\psi_{P,Q}}$$ are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of $${\psi_{P,Q}}$$ restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [] is improved. Properties of the functions $${\psi_{P,Q}}$$ are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of $${\psi_{P,Q}}$$ on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures $${\mu_1}$$ and $${\mu_2}$$ on $${{\mathbb{N}_2^\mathbb{N}}}$$ , we study which of these properties $${\psi_{P,Q}}$$ satisfies for $${\mu_1 \times \mu_2}$$ -almost every ( P, Q) $${{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}$$ . Related classes of random fractals are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider sets of points with some restricts on the digits of their α-Lüroth expansions. More precisely, for any countable partition α = { A, n ∈ ℕ} of the unit interval I, we completely determine the Hausdorff dimensions of the sets where φ is an arbitrary positive function defined on ℕ satisfying φ( n) → ∞ as n → ∞. [ABSTRACT FROM AUTHOR]

We consider the problem of slicing a compact metric space Ω with sets of the form $\pi_{\lambda}^{-1}\{t\}$, where the mappings π: Ω→ℝ, λ∈ℝ, are generalized projections, introduced by Yuval Peres and Wilhelm Schlag in 2000. The basic question is: Assuming that Ω has Hausdorff dimension strictly greater than one, what is the dimension of the 'typical' slice $\pi_{\lambda}^{-1}\{t\}$, as the parameters λ and t vary. In the special case of the mappings π being orthogonal projections restricted to a compact set Ω⊂ℝ, the problem dates back to a 1954 paper by Marstrand; he proved that for almost every λ there exist positively many t∈ℝ such that $\dim\pi_{\lambda }^{-1}\{t\} = \dim\varOmega- 1$. For generalized projections, the same result was obtained 50 years later by Järvenpää, Järvenpää and Niemelä. In this paper, we improve the previously existing estimates by replacing the phrase 'almost all λ' with a sharp bound for the dimension of the exceptional parameters. [ABSTRACT FROM AUTHOR]

Given a positive integer M and a real number $$q >1$$ , a q -expansion of a real number x is a sequence $$(c_i)=c_1c_2\ldots $$ with $$(c_i) \in \{0,\ldots ,M\}^\infty $$ such that It is well known that if $$q \in (1,M+1]$$ , then each $$x \in I_q:=\left[ 0,M/(q-1)\right] $$ has a q-expansion. Let $$\mathcal {U}=\mathcal {U}(M)$$ be the set of univoque bases $$q>1$$ for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of $$\mathcal {U}$$ and to show that the Hausdorff dimension of the set of numbers $$x \in I_q$$ with a unique q-expansion changes the most if q 'crosses' a univoque base. Denote by $$\mathcal {B}_2=\mathcal {B}_2(M)$$ the set of $$q \in (1,M+1]$$ such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741-754, 2009) and prove that where $$q'=q'(M)$$ is the Komornik-Loreti constant. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold: (1) we study different notions of the average distance between two points of a self-similar subset of $${\mathbb {R}}$$ , and (2) we investigate the asymptotic behaviour of higher order average moments of self-similar measures on self-similar subsets of $${\mathbb {R}}$$ . [ABSTRACT FROM AUTHOR]

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications. [ABSTRACT FROM AUTHOR]

We define and examine nonlinear potential by Bessel convolution with Bessel kernel. We investigate removable sets with respect to Laplace-Bessel inequality. By studying the maximal and fractional maximal measure, a Wolff type inequality is proved. Finally the relation of B-p capacity and B-Lipschitz mapping, and the B-p capacity and weighted Hausdorff measure and the B-p capacity of Cantor sets are examined. [ABSTRACT FROM AUTHOR]

We obtain new asymptotic results about systems of N particles governed by Riesz interactions involving k-nearest neighbors of each particle as N → ∞ . These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an N-body interaction. We find the first-order term of the large N asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the Γ -convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N. [ABSTRACT FROM AUTHOR]

In this paper we study a class of subsets of the general Sierpinski carpets for which two groups of allowed digits occur in the expansions with proportional frequency. We calculate the Hausdorff and Box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive and finite. [ABSTRACT FROM AUTHOR]

In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite. [ABSTRACT FROM AUTHOR]

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper. [ABSTRACT FROM AUTHOR]

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geometry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of σ-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reach) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, results on the absolute continuity of Hessian measures, and a duality theorem which relates the Hessian measures of a convex function to those of the conjugate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K. [ABSTRACT FROM AUTHOR]

In Ben Slimane (Mediterr J Math, 13(4):1513-1533 (2016)), the second author proved that, generically in the Baire category sense, pairs of functions in $${B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \times B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) }$$ , for $${s_{1} > \frac{m}{t_{1}}}$$ and $${s_{2} > \frac{m}{t_{2}}}$$ , satisfy a mixed multifractal formalism based on wavelet leaders. In this paper, we extend this validity on $${(B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \cap C^{\gamma_{1}}(\mathbb{R}^m)) \times (B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) \cap C^{\gamma_{2}}(\mathbb{R}^m)}$$ , for $${0 < \gamma_{1} < s_{1} < \frac{m}{t_{1}}}$$ and $${0 < \gamma_{2} < s_{2} < \frac{m}{t_{2}}}$$ . The main change is that the wavelet coefficients of the saturating function which generates the residual $${G_\delta}$$ set are not everywhere large enough and do not coincide everywhere with the wavelet leaders. Nevertheless, the computation of the wavelet leaders is done everywhere and allows to deduce both mixed spectra and mixed scaling function. [ABSTRACT FROM AUTHOR]

In the present work, we give an answer to questions posed in [3]. We precisely study the mutual singularity of Hewitt-Stromberg measures on Bedford-McMullen carpets. [ABSTRACT FROM AUTHOR]

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let be non-empty Borel sets. If X is not contained in any line, we prove that If dimHY>1, we have the following improved lower bound: Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck's theorem in combinatorial geometry: if is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines , then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}. While the results above concern , we also derive some counterparts in by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren. [ABSTRACT FROM AUTHOR]

In this paper, we show that the mixed Hölder spectra of finitely many functions can be recovered from a Legendre transform of a concave scaling function based on simultaneous wavelet leaders. We first prove that this Legendre transform yields an upper bound valid for uniform Hölder functions. We then study the typical optimality (in the sense of Baire's category) of this upper bound in a product of continuous Besov or oscillation spaces. [ABSTRACT FROM AUTHOR]

In this paper I construct and study new measures associated with a given pair of functions. To illustrate their applications in analysis, these measures are applied to the heat equation. [ABSTRACT FROM AUTHOR]

Ben Slimane (Math Proc Camb Philos Soc 124:329-363, ) has constructed specific anisotropic selfsimilar functions as counter-examples for the isotropic multifractal formalism. An anisotropic multifractal formalism has been formulated and its validity for anisotropic selfsimilar functions has been proved. In this paper, using Triebel anisotropic wavelet decompositions, we first obtain lower bounds of the anisotropic scaling function and upper bounds of the u-spectrum of singularities valid for all functions. We then investigate the generic validity, in the sense of Baire's categories, of the anisotropic formalism in some anisotropic functional spaces. We thus extend in the anisotropic setting some results of Jaffard (J Math Pure Appl 79:525-552, , Ann Appl Probab 10:313-329, ) and Jaffard and Meyer (Memoirs of the American Mathematical Society, vol. 123, ). [ABSTRACT FROM AUTHOR]

In this paper it is shown that if $$\mu $$ is a finite Radon measure in $${\mathbb R}^d$$ which is n-rectifiable and $$1\le p\le 2$$ , then where with the infimum taken over all the n-planes $$L\subset {\mathbb R}^d$$ . The $$\beta _{\mu ,p}^n$$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform n-rectifiability. An analogous necessary condition for n-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance $$W_1$$ is also proved. [ABSTRACT FROM AUTHOR]

In this paper, we extend the well-known Frostman lemma by showing that for any subset $$E$$ of $$[0, 1]$$ and $$\alpha >0$$ , if the $$\alpha $$ -Hausdorff measure of $$E$$ is positive, then there exist a non-zero Borel measure $$\mu $$ on $$[0, 1]$$ , a constant $$C>0$$ and a subset $$E_0$$ of $$E$$ such that $$\mu (I) \le C \vert I \vert ^{\alpha }$$ for any interval $$I$$ and $$E_0$$ is dense in the support of $$\mu $$ . Under an additional condition on $$E_0$$ , we show that $$\mu (B) = \mu [0, 1]$$ for any Borel subset $$B$$ containing $$E$$ . Using the notion of Choquet integral, we extend the notion of capacitarian dimension to arbitrary subset of $$[0, 1]$$ and prove a generalisation of Frostman's theorem. [ABSTRACT FROM AUTHOR]

Given a sequence { b i } i = 1 n and a ratio λ ∈ (0; 1); let E = ∪ i = 1 n (λ E + b i) be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in E. Our main idea is from the multiple β-expansions. [ABSTRACT FROM AUTHOR]

The aim of this work is to discuss the proportionality of the multifractal measures. We will prove that the ratio of the multifractal measures is bounded. In addition, for a class of homogeneous Cantor sets, we find an explicit formula for their multifractal Hausdorff and packing function dimensions and discuss some interesting examples. [ABSTRACT FROM AUTHOR]

We compute the centred Hausdorff measure, C s (P) ∼ 2.44 , and the packing measure, P s (P) ∼ 6.77 , of the penta-Sierpinski gasket, P , with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in P , which, in contrast with that of the Sierpinski gasket, consists of a unique interval. These results allow us to compute the irregularity index of P , I (P) ∼ 0.6398 , which we define for any self-similar set E with open set condition as I (E) = 1 - C s (E) P s (E) . [ABSTRACT FROM AUTHOR]

In this paper, an equivalent condition for the self similar sets on the real line to have best coverings is given. As a result, it partly gives answer to the conjecture which was posed by Zhou and Feng [Zhou, Z. L., Feng, L.: Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17(2), 493-502 (2004)]. [ABSTRACT FROM AUTHOR]