A nonlinear version of Roth's theorem for sets of positive density in the real line

It is proved that given ε>0, there is δ(ε)>0 such that ifS is a measurable set of [0,N], |S|>εN, then there is a triplex, x+h, x+h ² inS withh satisfyingh>δ(ε)N ^1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h ³,h ⁴ etc.     

Uniform approximation on compact subsets of the real line

Let {? _i } _i=∩ ⁿ be continuous real functions on the compact set M?R. We consider the problem of best uniform approximation of the function? by polynomials \(\sum\nolimits_{i = 1}^n {c_i \varphi _i }\) on M. Let V(?₀, A) be a set of polynomials of best approximation on A ? M. We show that \(V(\varphi _0 ,M) = \mathop \cap \limits_{A_{n + 1} } V(\varphi _0 ,A_{n + 1} )\) , where A_n+1 represents all the possible sets of n+ 1 points {x₁, ..., x_n+1} in M, containing the characteristic set of the given problem of best approximation and for which the the rank of ∥?_i ∥ (i=1, ...,n; j=1,..., n+1) is equal to n. This theorem is applied to a problem of uniform approximation where {? _i } _i=1 ⁿ is a weakly Chebyshev system.     

A note on the representability of compact sets by real sequences

Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l_∞( ). An approximation result is also proved.     

All compact manifolds are homeomorphic to totally algebraic real algebraic sets

Both authors supported in part by the N.S.F.     

Exceptional points for Lebesgue's density theorem on the real line

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither 0 nor 1. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities are closer to 1/2 than to zero or one. The method of proof uses a discretized restatement of the problem, and a self-similar construction.     

An equivalent condition for the self similar sets on the real line to have best coverings

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）].     

On the generalization of density topologies on the real line

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ? _J generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.     

Decomposing the real line into Borel sets closed under addition

We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on.     

On the density of integral sets with missing differences from sets related to arithmetic progressions

For a given set M of positive integers, a problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by an element of M. The problem is completely settled when |M|?2, and some partial results are known for several families of M for |M|?3, including the case where the elements of M are in arithmetic progression. We consider some cases when M either contains an arithmetic progression or is contained in an arithmetic progression.     

Product sets in the plane,sets of the formA+B on the real line and Hausdorff measures

Research supported by the Hungarian National Foundation for Scientific Research Grant No. 2114.     

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

Spaces of piecewise-continuous almost-periodic functions and almost-periodic sets on the real line. II

The authors consider a series of spaces of piecewise-continuous almost periodic functions and study the properties of the elements of these spaces. The theory developed in the paper is then applied to investigate almost periodic linear pulse systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 389–400, March, 1992.     

Compactness properties in the fuzzy real line

It is the purpose of this paper to give the solution to a number of open questions in [14], concerning the compactness properties of the fuzzy real line and the fuzzy real unit interval.However using the model of the fuzzy real line which we already proposed in [9, 12] we are able to describe all compactness properties not only of the spaces mentioned above but actually of several large classes of subspaces of the fuzzy real line; and thus the answers to the questions in [14] follow as easy consequences of our general results.     

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.     

Descriptive properties of density preserving autohomeomorphisms of the unit interval

We prove that density preserving homeomorphisms form a Π ₁ ¹ -complete subset in the Polish space ? of all increasing autohomeomorphisms of unit interval.     

Numerical inversion of the Mellin transform on the real line for heavy-tailed probability density functions

A method for numerical inversion on the real line of the Mellin transform, without reduction of the problem to the inversion of Laplace transform is described. Maximum entropy technique is invoked in choosing the analytical form of the approximant function. Entropy-convergence and then L₁-norm convergence is proved. A stability analysis in evaluating entropy and expected values is illustrated. An upper bound of the error in the expected values computation is provided in terms of entropy.