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It is proved that if a multiple series in the Franklin system converges in the sense of Pringsheim everywhere, except, perhaps, on a set that is a Cartesian product of sets of measure zero, to an everywhere finite integrable function, then it is the Fourier–Franklin series of this function. A uniqueness theorem is also proved for multiple Franklin series whose rectangular partial sums at each point have a sequential limit.

We study a norm sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet's theorem. Its supremum norm case was recently considered by the first-named author and Wadleigh, and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of a suitably scaled norm ball. We use methods from geometry of numbers and dynamics to generalize a result due to Andersen and Duke on measure zero and uncountability of the set of numbers for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface which visit a decreasing family of balls. An application of a dynamical Borel-Cantelli lemma of Maucourant produces a zero-one law for improvement of Dirichlet's theorem in Euclidean norm., Comment: 27 pages; Theorem 1.3 replaced by a stronger version, new section and more detail of the proof added

We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two., Comment: 42 pages. arXiv admin note: text overlap with arXiv:1505.07986

Let X be a normed space, Y be a Banach space and $$f,g: X\rightarrow Y$$. In this paper, we investigate the Hyers–Ulam stability theorem for the generalized quadratic functional equation $$\begin{aligned} f(kx+y)+f(kx-y)=2k^2g(x)+2f(y) \end{aligned}$$in a set $$\Omega \subset X\times X$$, where k is a positive integer. By the Baire category theorem, we derive some consequences of our main result.

Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on the real line cannot be decided in ZFC. We construct a matrix iteration of ccc posets to force that, for many ideals in that class, their associated cardinal invariants (i.e. additivity, covering, uniformity and cofinality) are pairwise different. In addition, we show that, consistently, the additivity and cofinality of Yorioka ideals does not coincide with the additivity and cofinality (respectively) of the ideal of Lebesgue measure zero subsets of the real line., Comment: 35 pages, 3 figures. Submitted

We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit interval [0, 1] infinitely. The set of points remained in the unit interval after this removal process is over is called the Cantor set. The dimension of such a set is not an integer value. In fact, it has a ‘fractional’ dimension, making it by definition a fractal. The Cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. In this paper, we have provided three types of generalization of the Cantor set depending on the process of removal. Also, we have discussed some characteristics of the fractal dimensions of these generalized Cantor sets. Further, we have shown its application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps.

Here we give an abstract formulation (in uncountable commutative groups) of a classical theorem of Sierpiński on the nonmeasurability of the algebraic sum of measure zero sets.

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

We show that for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (\emph{sub-Gaussian} estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M.\ T.\ Barlow in [Contemp.\ Math., vol.\ 338, 2003, pp.\ 11--40]. We also prove that in the contrary case of walk dimension two, i.e., where full off-diagonal \emph{Gaussian} estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form., Comment: 32 pages, 2 figures; several typos have been fixed (to appear in The Annals of Probability)

We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by J. Cima and R. Mortini. In particular we prove that for any inner function $\Theta$ whose singular set has measure zero, one can find a Blaschke product $B$ such that $\Theta B$ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions., Comment: 10 pages, 3 figures

We investigate strongly nonlinear differential equations of the type $$\bigl(\Phi \bigl(k(t) u'(t) \bigr) \bigr)'= f \bigl(t,u(t),u'(t) \bigr), \quad\text{a.e. on } [0,T], $$ where Φ is a strictly increasing homeomorphism and the nonnegative function k may vanish on a set of measure zero. By using the upper and lower solutions method, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions involving the function k. Our existence results require a weak form of a Wintner–Nagumo growth condition.

A new concept of small subsets of the real line is presented. It is a generalization of different kinds of microscopic sets considered previously. We compare these sets with Lebesgue null sets, with sets of strong measure zero. With this approach we are able to get families of fractals – sets with fractional Hausdorff dimension.

Ergodic homeomorphisms T and S of Polish probability spaces X and Y are evenly Kakutani equivalent if there is an orbit equivalence ϕ: X 0 → Y 0 between full measure subsets of X and Y such that, for some A ⊂ X 0 of positive measure, ϕ restricts to a measurable isomorphism of the induced systems T A and S ϕ(A). The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two zero-entropy loosely Bernoulli systems are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as the Morse minimal system are both measurable and topological. Recently del Junco, Rudolph and Weiss studied a new relation, called nearly continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an even Kakutani equivalence where also X 0 and Y 0 are invariant G δ sets, A is within measure zero of both open and closed, and ϕ is a homeomorphism from X 0 to Y 0. It is known that nearly continuous Kakutani equivalence is strictly stronger than even Kakutani equivalence, and nearly continuous Kakutani equivalence is the natural strengthening of even Kakutani equivalence to the nearly continuous category—the category of maps that are continuous after sets of measure zero are removed. In this paper, we show that the Morse minimal substitution system is nearly continuously Kakutani equivalent to the binary odometer.

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between $\mu$ and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an $L^2$ variant of P. Jones' traveling salesman construction, which is of independent interest., Comment: 47 pages, 4 figures (v3: added/updated figures, new Remarks 2.1, 4.6, 5.8, minor improvements, final version)

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood., 23 pages, 3 figures. We welcome any comments

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.

One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or H\"older or $C^{1,\alpha}$ images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro. This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017., Comment: 28 pages, 3 figures (v2: corrected typos and updated references, final version)

We formulate and prove a Gelfand-Levitan trace formula for general quantum graphs with arbitrary edge lengths and coupling conditions which cover all self-adjoint operators on quantum graphs, except for a set of measure zero. The formula is reminiscent of the original Gelfand-Levitan result on the segment with Neumann boundary conditions., 16 pages. This is a preprint of an article published in Anal. Math. Phys. The final authenticated version is available online at: https://doi.org/10.1007/s13324-021-00487-3

In this chapter, we discuss the Hyers–Ulam stability theorem for the σ-Drygas functional equation $$\displaystyle f(x+y)+f\big (x+\sigma (y)\big )=2f(x)+f(y)+f\big (\sigma (y)\big ) $$ for all \((x,y)\in \varOmega \subset \mathbb {C}^{2}\) for Lebesgue measure m(Ω) = 0, where \(f:\mathbb {C}\to Y\) and σ : X → X is an involution.

In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $\sigma$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^\omega$, $\mathbb R^\omega$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product., Comment: 19 pages

We introduce and study a continuous function I which is called a digit inversor for the Q 3-representation of the fractional part of a real number. This representation is determined by a probability vector.(q 0; q 1; q 2) with positive coordinates, generalizes the classical ternary representation, and coincides with this representation for q 0 = q 1 = q 2 = 1/3: The values of this function are obtained from the Q 3-representation of the argument by the following change of digits: 0 by 2; 1 by 1; and 2 by 0: The differential, integral, and fractal properties of the inversor are described. We prove that I is a singular function for q 0 ≠ q 2.

We show the existence of Levy-type stochastic processes in one space dimension with characteristic triplets that are either discontinuous at thresholds, or are stable-like with stability index functions for which the closures of the discontinuity sets are countable. For this purpose, we formulate the problem in terms of a Skorokhod-space martingale problem associated with non-local operators with discontinuous coefficients. These operators are approximated along a sequence of smooth non-local operators giving rise to Feller processes with uniformly controlled symbols. They converge uniformly outside of increasingly smaller neighborhoods of a Lebesgue null set on which the singularities of the limit operator are located.

The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.

We develop a theory of \emph{sharp measure zero} sets that parallels Borel's \emph{strong measure zero}, and prove a theorem analogous to Galvin-Myscielski-Solovay Theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^\omega$ is meager-additive if and only if it is $\mathcal E$-additive; if $f:2^\omega\to2^\omega$ is continuous and $X$ is meager-additive, then so is $f(X)$., Comment: arXiv admin note: text overlap with arXiv:1208.5521

We construct the Lebesgue integral in three steps. First for simple functions, then for positive measurable functions, and finally for all real valued measurable functions. The more technical work is done for simple functions and is then used for the more general sets of functions.

We show that every homeomorphism between closed measure zero subsets extends to a measure preserving auto-homeomorphism, whenever the Cantor set is endowed with a suitable probability measure. This is valid both for the standard product measure, as well as for the universal homogeneous rational measure., Comment: Some corrections, new subsection added, final version (18 pages)

Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers. The most basic question is: how hard is it to compute a finite sub-cover from such a cover of $2^{\mathbb{N}}$? Another natural question is: how hard is it to compute a sequence that covers $2^{\mathbb{N}}$ minus a measure zero set from such a cover? The special and weak fan functionals respectively compute such finite sub-covers and sequences. In this paper, we establish the connection between these new fan functionals on one hand, and various well-known comprehension axioms on the other hand, including arithmetical comprehension, transfinite recursion, and the Suslin functional. In the spirit of Reverse Mathematics, we also analyse the logical strength of compactness in Nonstandard Analysis. Perhaps surprisingly, the results in the latter mirror (often perfectly) the computational properties of the special and weak fan functionals. In particular, we show that compactness (nonstandard or otherwise) readily brings us to the outer edges of Reverse Mathematics (namely $\Pi_2^1$-CA$_0$), and even into Schweber's higher-order framework (namely $\Sigma_{1}^{2}$-separation)., Comment: To appear in Annals of Pure and Applied Logic

Lebesgue measure on R n is restriction of outer Lebesgue measure to the family of Lebesgue measurable sets. All properties of Lebesgue positive measure on R are transfered to Lebesgue measure on R n . The Lebesgue measure on B is only meassure defined on σ-algebra of Borel sets of R n , which is invariance to translation and .

We provide sufficient conditions for a set $E\subset\mathbb{R}^n$ to be a non-universal differentiability set, i.e. to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of $\mathbb{R}^n$ given by Alberti, Cs��rnyei and Preiss, which eventually led to the result of Jones and Cs��rnyei that for every Lebesgue null set $E$ in $\mathbb{R}^n$ there is a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^n$ not differentiable at any point of $E$, even though for $n>1$ and for Lipschitz functions from $\mathbb{R}^n$ to $\mathbb{R}$ there exist Lebesgue null universal differentiability sets., 30 pages

Given $n\in\mathbb{N}$ and $\tau>\frac1n$, let $\mathcal{S}_n(\tau)$ denote the classical set of $\tau$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-\tau-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau>\frac1n$ almost all points on $\mathcal{M}$ are not $\tau$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set $\mathcal{M}\cap\mathcal{S}_n(\tau)$. In this paper we suggest a new approach based on the Mass Transference Principle, which enables us to find a sharp lower bound for $\dim \mathcal{M}\cap\mathcal{S}_n(\tau)$ for any $C^2$ submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $\tau$ satisfying $\frac1n\le\tau, Comment: 20 pages

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma$-porous with respect to the Carnot-Carath\'eodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups we then construct a Lipschitz function which is Pansu differentiable at no point of a given $\sigma$-porous set and show preimages of open sets under the horizontal gradient are far from being porous., Comment: 23 pages

Let R be the set of real numbers, Y a Banach space and f : R → Y . We prove the Hyers–Ulam stability theorem for the quadratic functional inequality ‖ f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) ‖ ≤ ϵ for all ( x , y ) ∈ Ω , where Ω ⊂ R 2 is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.

We show that the set of all separable Banach spaces that have the π-property is a Borel subset of the set of all closed subspaces of C(Δ), where Δ is the Cantor set, equipped with the standard Effros-Borel structure. We show that if α < ω 1, the set of spaces with Szlenk index at most α which have a shrinking FDD is Borel.

We prove that, for general cost functions on R n , or for the cost d 2/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.

We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side., Comment: Final version, to appear in Comm. Pure Appl. Math

This paper is concerned with weak⁎ closed masa-bimodules generated by A ( G ) -invariant subspaces of VN ( G ) . An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B ( L 2 ( G ) ) which are invariant under both Schur multipliers and a canonical action of M ( G ) on B ( L 2 ( G ) ) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

We give in \mathbb{R}^6 a real analytic almost complex structure J, a real analytic hypersurface M and a vector v in the Levi null set at 0 of M, such that there is no germ of J-holomorphic disc f included in M with f(0)=0 and \frac{\partial f}{\partial x}(0)=v, although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J, we give necessary conditions under which there exists such a germ of disc.

In this note we describe some geometrical properties that simplicially convex bodies typically enjoy. It is shown, for example, that they are nowhere dense and of measure zero. Moreover, they look at least half-dense from any of their points.

[EN] We prove that the set of points doubly asymptotic to a point has measure zero with respect to any expansive outer regular measure for a bi-measurable map on a separable uniform space. Consequently, we give a class of measures which cannot be expansive for Denjoy home-omorphisms on S1. We then investigate the existence of expansive measures for maps with various dynamical notions. We further show that measure expansive (strong measure expansive) homeomorphisms with shadowing have periodic (strong periodic) shadowing. We relate local weak specification and periodic shadowing for strong measure expansive systems., The first author is supported by the Department of Science and Technology, Government of India, under INSPIRE Fellowship (Registration No- IF150210) Program since March 2015.

We develop a Fourier-Feynman theory for Fourier-type func-tionals k F and [F on Wiener space. We show that Fourier-Feynmantransform and convolution of Fourier-type functionals exist. We alsoshow that the Fourier-Feynman transform of the convolution product ofFourier-type functionals is a product of Fourier-Feynman transforms ofeach functionals. 1. Introduction and preliminariesLet C 0 [0;T] denote the Wiener space, that is, the space of real valuedcontinuous functions xon [0;T] with x(0) = 0. Let Mdenote the class ofall Wiener measurable subsets of C 0 [0;T] and let mdenote Wiener measure.(C 0 [0;T];M;m) is a complete measure space and we denote the Wiener integralof a functional FbyZ C 0 [0;T] F(x)dm(x):A subset Eof C 0 [0;T] is said to be scale-invariant measurable provided ˆEis Wiener measurable for every ˆ>0, and a scale-invariant measurable setN is said to be scale-invariant null provided m(ˆN) = 0 for every ˆ>0. Aproperty that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). Given two complex-valued functions Fand Gon C

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

Let E is a complex vector space and F is real (or complex ) Banach space. In this paper, we prove the Hyers-Ulam stability for the generalized Jensen functional equation .

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an example of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect., 14 pages. See also http://rstankewitz.iweb.bsu.edu/, http://sugawa.cajpn.org/index_E.html, http://www.math.sci.osaka-u.ac.jp/~sumi/

We add to the literature the following observation. If µ is a singular measure on the real line which assigns measure zero to every porous set and f : R ! R is a Lipschitz function which is non-differentiable µ-a.e. then for every C 1 function g : R ! R there holds µ{x 2 R : f(x) = g(x)} = 0. In other words the Lusin type approximation property of Lipschitz functions with C 1 functions does not hold with respect to a general Radon measure. Moreover we discuss, only in the one-dimensional setting, how the result contained in (Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991)) could be extended and improved when the Lebesgue measure is replaced by an arbitrary Radon measure.

Singular quadratic forms are encountered in numerous mathematical and physical problems. Thus, in mathematical physics, the investigation of singular potentials is connected with the analysis of formal mathematical expressions which are meaningful only as quadratic forms. In model problems of quantum physics, some requirements (such as relativistic invariance or infiniteness of the number of degrees of freedom) necessarily lead to the appearance of quadratic forms connected with generalized functions. A large class of singular forms arises in the course of investigation of so-called potentials with small supports, i.e., potentials concentrated on sets of measure zero (see, e.g., [AdH], [AdP], [AGHKH], [AFHKKL], [AHKS1], [AHKS2]. [A1K2], [A1K3], [AGS], [Ara], [BeP], [Bra2], [BEKS], [BrT], [Cheb], [Chr], [Chu], [Dan], [DFT1], [DeO], [Hed], [Hep], [HKHJWL], [Jos], [KaM], [KrP], [Nel], [PeG], [Por], [She], and [TuP]).

In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.

J.C. Oxtoby and S. Ulam proved that each set of the first category in r -dimensional Euclidean space can be transformed into a set of Lebesgue measure zero by some automorphism, and such homeomorphisms constitute a residual set in the set of all automorphisms. We have improved this result changing the sets of measure zero by microscopic sets on the real line and on the plane.