Let A be a sequence of positive integers and P(A) be the set of all integers which are the finite sum of distinct terms of A. Let \(B=\{b_1

The generalized topology on the Cartesian product of sets can be defined by the generalized topology on the factors of the product. A G-method on a set can derive a generalized topology, which is called a G-generalized topology. On the one hand, we introduce the concept of product G-methods on sets which lead to a G-generalized topology on the Cartesian products that is different both from the Császár

The notion of \(\mu\)-monotonically T2 spaces is introduced, which is a generation of monotone T2 separation in general topological spaces. The characterization and some properties of \(\mu\)-monotonically T2 spaces are given. Besides, the definition of the box product of the topologies is extended to generalized topologies, and we prove that the box product of \(\mu\)-monotonically T2 spaces is \

An example is constructed of a subset \(A \subset \mathbb {Z}_N\) of density \({1\over 2}+o(1)\) such that all the non-trivial Fourier coefficients of the characteristic function of A are very small, but if \(x,d \in \mathbb {Z}_N\) are chosen uniformly at random, then the probability that \(x, x+d, x+2d\) and \(x+3d\) all belong to A is at most \({1\over 16}-c\), where \(c>0\) is an absolute constant

The set A is an asymptotic nonbasis of order h for an additive abelian semigroup X if there are infinitely many elements of X not in the h-fold sumset hA. For all \(h \geq 2\), this paper constructs new classes of asymptotic nonbases of order h for Z and for N0 that are not subsets of maximal asymptotic nonbases.

For \(k \in \mathbb {N}\), write S(k) for the largest natural number such that there is a k-colouring of \(\{1, \ldots ,S(k)\}\) with no monochromatic solution to \(x-y=z^2\). That S(k) exists is a result of Bergelson, and a simple example shows that \(S(k) \ge 2^{2^{k-1}}\). The purpose of this note is to show that \(S(k) \le 2^{2^{2^{O(k)}}}\) .

It is well known that for every even integer n, the complete graph \(K_{n}\) has a one-factorization, namely a proper edge coloring with \(n-1\) colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of large one-factorizations. In particular, we know essentially nothing about the typical properties

Let A be a finite subset of \(\mathbb{Z}^n\), which generates \(\mathbb{Z}^n\) additively. We provide a precise description of the N-fold sumsets NA for N sufficiently large, with some explicit bounds on “sufficiently large.”

We show that every 4-uniform hypergraph with n vertices and minimum pair degree at least \((5/9+o(1))n^{2}/2\) contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.

We investigate 4-chromatic Schrijver graphs from various points of view and color-critical edges in Schrijver graphs in general. In particular, we present the following results. We give an elementary proof for the non-3-colorability of 4-chromatic Schrijver graphs thus providing such a proof also for 4-chromatic Kneser graphs. We show that only certain types of edges of Schrijver graphs can be color-critical

We show that the de Bruijn–Erdős condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence \(0\leq f(1)\leq f(2)\leq f(3) \leq \cdots\) such that $$(1) \qquad \qquad \sum_{ n=1}^{\infty} f(n)/n^2 = \infty. $$ Then, there exists a sequence \(\{b(n)\}_{n=1,2,\ldots }\) satisfying $$(2) \qquad

We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa–Szemerédi graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemerédi. The graphs induced by this protocol have n vertices, \(\Omega(n^2/\log n)\) edges, and are decomposable into \(n^{1+O(1/\log \log n)}\) induced matchings. Another protocol is a somewhat simpler version

If we want to color \(1,2,\ldots ,n\) with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least n/2 colors. Surprisingly, much fewer colors suffice if we are allowed to leave a negligible proportion of integers uncolored. Specifically, we prove that there exist \(\alpha ,\beta <1\) such that for

Szemerédi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems

In 1975, Szemerédi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemerédi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting

Linnik considered about 70 years ago the following approximation to the binary Goldbach problem. Is it possible to give a fixed integer K such that every sufficiently large even integer could be written as the sum of two primes and K powers of two? He solved the problem affirmatively with an unspecified large K. The first explicit result \((K=54\,000)\) appeared at the end of the 1990's. In the present

There is a result, due independently to Kurepa [14] and to Jurkat [12], which distinguishes linear functions or derivations from other additive functions as solutions to certain functional equations. The purpose of this paper is to prove an analogue of a part of this result, corresponding to derivations, for quadratic functions.

For set \(A\subset \mathbb{F}_p^*\) define by \(\mathsf{sf}(A)\) the size of the largest sum-free subset of A. Alon and Kleitman [3] showed that \(\mathsf{sf} (A) \ge |A|/3+O(|A|/p)\). We prove that if \(\mathsf{sf}(A)-|A|/3\) is small then the set A must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups

Let \(\mathbb{Z}_{n}\) be the ring of residue classes modulo n, \(\mathbb{Z}_{n}^{*}\) be its unit group, and let \(f(z)=az^{2}+bz\) be an integral quadratic polynomial with \(b\neq0\) and \({\rm gcd}(a,b)=1\). In this paper, for any integer c and positive integer n we give a formula for the number of solutions of the congruence equation \(f(x)+f(y)\equiv c({\rm mod}\, n)\) with x, y units \(({\rm

We introduce the basic concepts of spherical spectral synthesis on hypergroups, following the ideas of our paper [11]. The commutativity of the hypergroup X is not assumed, we suppose only that (X,K) is a Gelfand pair, that is, K is a compact subhypergroup in X and the convolution algebra of allK-invariant compactly supported complex Borel measures on X is commutative.

A set of m distinct positive integers \(\{a_{1} , \ldots a_{m}\}\) is called a \(D(q)-m\)-tuple for nonzero integer q if the product of any two increased by q, \(a_{i}a_{j}+q, i \neq j\) is a perfect square. Due to certain properties of the sequence, there are many D(q)-Diophantine triples related to the Fibonacci numbers. A result of Baćić and Filipin characterizes the solutions of Pellian equations

The main goal of this paper is to extend Flanigan’s theorem in [4] concerning ideal essential extensions of rings to left ideal essential extensions. Moreover, we give new proofs of two Flanigan’s theorems and answer a question raised by M. Petrich in [5] which is related to pure extensions of rings.

Let \(d_{(1)}(n)\) be the n-th coefficient of the Dirichlet series \((\zeta '(s))^{2}=\sum _{n=1}^{\infty }d_{(1)}(n)n^{-s}\) in \(\mathfrak {R}s>1\), and \(\Delta _{(1)}(x)\) be the error term of the sum \(\sum _{n\le x}d_{(1)}(n)\). In this paper, we study the higher power moments of \(\Delta _{(1)}(x)\) and derive asymptotic formulas for $$\int _{1}^{T}\Delta _{(1)}^{k}(x)\, dx , \quad k=3,\ldots

There are several important separation theorems in various areas; for example, theorems of Gordan, Stone, Mazur, Hahn–Banach, etc. In this paper, we give a general treatment, in ZFC, of separation results with several examples in old and new settings. In order to achieve some uniformity of the treatment, we define the notion of a solid operator that leads to the notion of separation. We also characterize

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey’s Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

The aim of this paper is to give linear independence results for the values of Lambert type series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences \(\{n_\ell\}_{\ell\geq1}\). For example, the three numbers $$1, \quad \sum_{p:{\rm prime}}\frac{1}{F_{p^2}}, \quad \sum_{p:{\rm prime}}\frac{1}{L_{p^2}}

We exhibit an analogy between the problem of pushing forward measurable sets under measure preserving maps and linear relaxations in combinatorial optimization. We show how invariance of hyperfiniteness of graphings under local isomorphism can be reformulated as an infinite version of a natural combinatorial optimization problem, and how one can prove it by extending wellknown proof techniques (linear

Let \(F_{n}\) and \(L_{n}\) be the nth Fibonacci and Lucas number, respectively. The order of appearance is defined as the smallest natural number k such that n divides \(F_{k}\) and denoted by z(n) . In this paper, we give explicit formulas for the terms \( z(F_{a}F_{b}) \), \( z( L_{a}L_{b}) \), \( z(F_{a}L_{b}) \) and \( z(F_{n}F_{n+p}F_{n+2p}) \) with \(p\ge 3\) prime.

We obtain an optimal upper bound for the normalised volume of a hyperplane section of an origin-symmetric d-dimensional cube. This confirms a conjecture posed by Imre Bárány and Péter Frankl.

Let \(p(\cdot ) \mathbb{T}\it ^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder condition and \(0

An edge coloring of a graph is a local r-coloring if the edges incident to any vertex are colored with at most r distinct colors. In this paper, generalizing our earlier work, we study the following problem. Given a family of graphs \(\mathcal {F} \) (for example matchings, paths, cycles, powers of cycles and paths, connected subgraphs) and fixed positive integers s, r, at least how many vertices can

We introduce a new version of characterized subgroups of the circle group \(\mathbb{T}\) that we call "\(\alpha\)-statistically characterized subgroups", in line of the very recent work [11], using the notion of statistical convergence of order \(\alpha\) [4]. We show that for any arithmetic sequence \((a_n)\) and \(\alpha \in (0, 1)\), the \(\alpha\)-statistically characterized subgroups \(t^\alp

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who

We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form $$\begin{aligned} \int _{\{ {x: \mathcal {M} f(x) > \lambda } \}} {\varphi (\lambda {\omega _1}(x))} {\omega _2}(x) \,d\mu \le {c} \int _X \varphi ({c}f(x){\omega _3}(x)){\omega _4}(x) \,d\mu \end{aligned}$$ in Orlicz classes, where \(\mathcal {M} f\) is a Hardy–Littlewood maximal function

We present various properties of the Erlang loss function $$B(x,a)=\Bigl( a \int_0^\infty e^{-at} (1+t)^x \,dt \Bigr) ^{-1} \quad{(x\geq 0,\ a>0)}.$$ Among other results, we prove: (1) The function \(x\mapsto B(x,a)^{\lambda}\) is convex on \([0,\infty)\) for every \(a>0\) if and only if \(\lambda\leq 0\) or \(\lambda \geq 1\). (2) The function \(x\mapsto ({1-B(1/x,a)})^{-1} \) is strictly convex on

The optimality of the Erdős–Rado theorem for pairs is witnessed by the colouring \(\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa\) recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which \(\Delta_\kappa\) is an extremal such triangle-free or

Previously, Erdős, Kierstead and Trotter [5] investigated the dimension of random height 2 partially ordered sets. Their research was motivated primarily by two goals: (1) analyzing the relative tightness of the Füredi–Kahn upper bounds on dimension in terms of maximum degree; and (2) developing machinery for estimating the expected dimension of a random labeled poset on n points. For these reasons

We generalize an important property of trigonometric series to the case of series by orthonormal spline systems corresponding to the dyadic sequence of grid points. We prove that Ciesielski series cannot diverge to infinity on a set of positive measure.

We consider countable edge colourings of complete graphs and investigate the existence of monochromatic paths. We show how this depends on the sizes of the complete graphs whose edges are being coloured.

We prove two theorems showing connections between multipliers for the symmetrized Jacobi expansions and multipliers for the Dunkl transform on \(\mathbb{R}\). These results can be seen as generalizations of the classical Igari’s result relating Jacobi and Hankel multipliers. Moreover, we show a transference theorem relating Jacobi and Dunkl transplantations.

We study integral operators with kernels $$K(x,y)= k_{1}( x- A_1y) \cdots k_{m}( x-A_my),$$ \(k_{i}(x)=\frac{\Omega_{i}(x)}{|x|^{n/q_i}}\) where \(\Omega_{i} \colon \mathbb{R}^{n} \to \mathbb{R}\) are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and \(\frac n{q_1}+\cdots+ \frac n{q_m} = n - \alpha, 0 \leq \alpha

We introduce the concepts of p-proximal contraction and p-proximal contractive mappings on metric spaces. Then we give some best proximity point results for such mappings. Also we provide some illustrative examples to compare our results with some earliers.

We find all the solutions of the title Diophantine equation \(P_1^p+2P_2^p + \cdots +kP_k^p=P_n^q\) in positive integer variables \((k, n)\), where \(P_i\) is the \(i^{th}\) term of the Pell sequence if the exponents p, q are included in the set \(\{1,2\}\).

We introduce a definition of \({\pi}\) being injective with respect to a generalized topology and a hereditary class where \({\pi}\) is a generalized quotient map between generalized topological spaces. This definition is mainly a sufficient condition to show several relations about a generalized topology and its induced generalized quotient topology when either is extended by a hereditary class or

We prove that if the sequence \(\langle k_n:1 \le n < \omega\rangle\) contains a so-called gap then the sequence \(\langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle\) of stationary sets is not mutually stationary, provided that \(k_n

We give a regular extension of the Menon-type identity to residually finite Dedekind domains.

We extend the notion of Aczél’s iterations of a mean to three-variable means and prove that these iterations can be used to define a continuous and scale invariant weigthing procedure on an extensive class of means.

Let n, k and a be positive integers. The Stirling numbers of the first kind, denoted by s(n, k), count the number of permutations of n elements with k disjoint cycles. Let p be a prime. Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the p-adic valuations of s(n, k). In this paper, by using Washington’s congruence on the generalized harmonic

Let \(\beta >1\) be a non-integer. First we show that Lebesgue almost every number has a \(\beta \)-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many \(\beta \)-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced \(\beta \)-expansions, where an infinite sequence on the finite alphabet \(\{0

This birthday note gives a “non-asymptotic” version of our earlier result with G. N. Sárközy and Szemerédi [3], in which Endre had the lion’s share. A hypergraph H with vertex set V defines the shadow graph G(H) whose vertex set is V and whose edge set is the set of pairs of V that are covered by some hyperedge of H. An edge coloring C of H defines a multicoloring, the shadow coloring\(C^{\prime}\)

As it was introduced by Tkachuk and Wilson in [7], a topological space X is cellular-compact if for any cellular, i.e. disjoint, family \(\mathcal{U}\) of non-empty open subsets of X there is a compact subspace \(K \subset X\) such that \(K \cap U \ne \emptyset\) for each \(U \in \mathcal{U}\). In this note we answer several questions raised in [7] by showing that (1) any first countable cellular-compact

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces.

We pursue the investigation of the concept of n-distance, ann-variable version of the classical concept of distance recently introduced and investigated by Kiss, Marichal, and Teheux. We especially focus on the challenging problem of computing the best constant associated with a given n-distance. In particular, we define and investigate the best constants related to partial simplex inequalities. We

It is proved that the maximal operator of subsequences of the Cesàro and Riesz means with varying parameters is bounded from the dyadic Hardy space Hp to Lp. This implies an almost everywhere convergence for the subsequences of tqoo.

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot hold in full generality. In the present paper, we formulate

We investigate the behavior of Fourier coefficients of Lip α class functions with respect to general orthonormal systems (ONS). The generalizations in a certain sense of some theorems by S. Szász, B. I. Golubov and S. V. Bochkarev are obtained.

The theory of Uniform Distribution started with the equidistribution of the irrational rotation of the circle, proved around 1905 independently by Bohl, Sierpinski and Weyl. The quantitative upgrading of this qualitative result was done by Hardy, Littlewood, Ostrowski, Weyl, and others around 1920. Their work pointed out the crucial role of continued fractions in uniform distribution (for a good treatment

Let X, Y be sets and let \(\Phi, \Psi\) be mappings with the domains X2 and Y2 respectively. We say that \(\Phi\) is combinatorially similar to \(\Psi\) if there are bijections \(f \colon \Phi(X^2) \to \Psi(Y^{2})\) and \(g \colon Y \to X\) such that \(\Psi(x, y) = f(\Phi(g(x), g(y)))\) for all \(x, y \in Y\). It is shown that the semigroups of binary relations generated by sets \(\{\Phi^{-1}(a) \colon

We show that, under natural conditions, the minimal, respectively maximal tensor product of a pro-C*-algebra associated to a pro-C*-correspondence and a pro-C*-algebra is isomorphic to the pro-C*-algebra associated with a tensor product pro-C*-correspondence.

We introduce the perturbed version of the Barabási–Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation, then the resulting degree distribution will