Lower bounds for the number of hyperplanes separating two finite sets of points

We consider the NP-hard problem of polyhedral separability of two finite sets A and B of points in general position in ? ^d by the minimum number of hyperplanes in the sense of a boolean function from a given class Σ. Both deterministic and probabilistic lower bounds are obtained for this number for two different classes of functions Σ.     

On the irreducible components of globally defined semianalytic sets

In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable C-semianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

On the index sets of Σ-subsets of the real numbers

We compute the levels of complexity in analytical and arithmetical hierarchies for the sets of the Σ-formulas defining in the hereditarily finite superstructure over the ordered field of the reals the classes of open, closed, clopen, nowhere dense, dense subsets of ? ⁿ , first category subsets in ? ⁿ as well as the sets of pairs of Σ-formulas corresponding to the relations of set equality and inclusion which are defined by them. It is also shown that the complexity of the set of the Σ-formulas defining connected sets is at least Π ₁ ¹ .     

Sumsets of dense sets and sparse sets

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B:= «a+b: a ∈ A, b ∈ B» is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

Symmetric complete sum-free sets in cyclic groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0, 1/3], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of ?_p of size at least (1/3?c)·p, where c > 0 is a universal constant.     

Quotient and product sets of thin subsets of the positive integers

We study the cardinalities of A/A and AA for thin subsets A of the set of the first n positive integers. In particular, we consider the typical size of these quantities for random sets A of zero density and compare them with the sizes of A/A and AA for subsets of the shifted primes and the set of sums of two integral squares.     

Counting sets with small sumset and applications

We study the number of k-element sets A? {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2 ^{o ( k )} N ^{o (1)} for most N and k. As a consequence of this and a further new result concerning the number of sets A??/N? with |A+A| ≤ c|A|², we deduce that the random Cayley graph on ?/N? with edge density ½ has no clique or independent set of size greater than (2+o(1)) log₂ N, asymptotically the same as for the Erd?s-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log₂ N was obtained. As a second application, we show that if the elements of A ? ? are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ? is equal to (2+O(1))^?k/2 as k → ∞.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

Some dimensional results for a class of special homogeneous Moran sets

We construct a class of special homogeneous Moran sets, called {m_k}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {m_k}_k?1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works.     

On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple (x, y, z) of not necessarily distinct elements with x + y = z. In this paper, we provide a structural characterisation of sum-free subsets of {1, 2,..., n} of density at least 2/5 ? c, where c is an absolute positive constant. As an application, we derive a stability version of Hu’s Theorem [Proc. Amer. Math. Soc. 80 (1980), 711–712] about the maximum size of a union of two sum-free sets in {1, 2,..., n}. We then use this result to show that the number of subsets of {1, 2,..., n} which can be partitioned into two sum-free sets is Θ(2^4n/5), confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].     

Gabor pairs,and a discrete approach to wave-front sets

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of “continuous type”. This implies that the coefficients of a Gabor frame expansion of f are parameter dependent, and describe the wave-front set of f.     

Generalized continuous functions defined by generalized open sets on generalized topological spaces

We introduce generalized continuous functions defined by generalized open (= g-α-open, g-semi-open, g-preopen, g-β-open) sets in generalized topological spaces which are generalized (g, g′)-continuous functions. We investigate characterizations and relationships among such functions.     

Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets.     

Nonlinear functions and difference sets on group actions

There are many generalizations of the classical Boolean bent functions. Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They are generalizations of perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X in terms of a (G, H)-related difference family. When G is abelian, we prove that there is a normalized G-dual set \(\widehat{X}\) of X, and characterize a G-difference set of X by the Fourier transform on a normalized G-dual set \({{\widehat{X}}}\). We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, 2001; Des Codes Cryptogr 46:83–96, 2008; GESTS Int Trans Comput Sci Eng 12:1–14, 2005; Linear Algebra Appl 452:89–105, 2014) are direct consequences of our results.     

Finite configurations in sparse sets

Let E ? ?ⁿ be a closed set of Hausdorff dimension α. For m > n, let{B₁, …, B_k} be n × (m ? n) matrices. We prove that if the system of matrices B_j is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translate of a non-trivial k-point configuration {B₁y, …, B_ky}. As a consequence, we are able to establish existence of certain geometric configurations in Salem sets (such as parallelograms in ?ⁿ and isosceles right triangles in ?²). This can be viewed as a multidimensional analogue of the result of [25] on 3-term arithmetic progressions in subsets of ?.     

A generalization of the notion of microscopic sets

We investigate families of subsets of the real line defined by nonincreasing sequences of positive real numbers. One of these families coincides with the σ-ideal of microscopic sets. We prove that the union of our families is equal to the σ-ideal of Lebesgue measure zero sets and the intersection of all such families is the σ-ideal of sets of strong measure zero. We also study other properties concerning homeomorphisms between sets of the first category and sets from our families.