Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

On a method of Holopainen and Rickman

There exists a quasiregular map on ℝ ⁿ (n≥3) of finite order for which every ℝ ⁿ is an asymptotic value.     

Smooth Quasiregular Maps with Branching in R n

According to a theorem of Martio, Rickman and Väisälä, all nonconstant C^n/(n-2)-smooth quasiregular maps in Rⁿ, n≥3, are local homeomorphisms. Bonk and Heinonen proved that the order of smoothness is sharp in R³. We prove that the order of smoothness is sharp in R⁴. For each n≥5 we construct a C^1+ε(n)-smooth quasiregular map in Rⁿ with nonempty branch set.     

Sieve-equivalence and explicit bijections

Suppose A₁,…, A_n are subsets of a finite set A, and B₁,…, B_n are subsets of a finite set B. For each subset S of N = {1, 2,…, n}, let A_s = ∩_i?SA_i and B_S = ∩_i?SB_i. It is shown that if explicit bijections f_S:A_S → B_S for each S ? N are given, an explicit bijection h:A-∪_i=1A_i→B-∪_i=1B_i can be constructed. The map h is independent of any ordering of the elements of A and B, and of the order in which the subsets A_i and B_i are listed.     

Cofinite subsets of asymptotic bases for the positive integers

Let A be an infinite set of integers containing at most finitely many negative terms. Let hA denote the set of all integers n such that n is a sum of h elements of A. Let F be a finite subset of A.Theorem. If hA contains an infinite arithmetic progression with difference d, and if $\text{gcd}\text{{a?a′¦a,a′∈AF} = d}$, then there exists q such that q(A\F) contains an infinite arithmetic progression. In particular, if A is an asymptotic basis, then A\F is an asymptotic basis if and only if gcd{a?a′|a,a′∈A\F} = 1.Theorem. If A is an asymptotic basis of order h, and if F?A, card(F) = k, and A\F is an asymptotic basis, then the exact order of A\F is O(h^k+1).     

On extendibility of unavoidable sets

A subset X of a free monoid A¹ is said to be unavoidable if all but finitely many words in A¹ contain some word of X as a subword. A. Ehrenfeucht has conjectured that every unavoidable set X is extendible in the sense that there exist x ? X and a ? A such that (X ? {x}) ∪ {xa} is itself unavoidable. This problem remains open, we give some partial solutions and show how to efficiently test unavoidability, extendibility and other properties of X related to the problem.     

On a characterization of Arakelian sets

Let K be a compact set in the complex plane, such that its complement in the Riemann sphere, (? ∪ {∞}) / K, is connected. Also, let U ? ? be an open set which contains K. Then there exists a simply connected open set V ? ? such that K ? V ? U. We show that if K is replaced by a closed set F ? ?, then the preceding result is equivalent to the fact that F is an Arakelian set in ?. This holds in more general case when ? is replaced by any simply connected open set Ω ? ?. In the case of an arbitrary open set Ω ? ?, the above extends to the one point compactification of Ω. If we do not require (? ∪ {∞}) /K to be connected, we can demand that each component of (? ∪ {∞}) / V intersects a prescribed set A containing one point in each component of (? ∪ {∞}) / K. Using the previous result, we prove that again if we replace K by a closed set F, the latter is equivalent to the fact that F is a set of uniform meromorphic approximation with poles lying entirely in A.     

Dense minimal asymptotic bases of order two

We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erd?s and Nathanson showed that for every h?2 there exists a minimal asymptotic basis A of order h with d(A)=1/h, where d(A) denotes the density of A. Erd?s and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order h?2 such that A(k)=k/h+O(1). Moreover, they asked if there exists a minimal asymptotic basis with lim sup(ai+1−ai)=3. In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition

     

Piecewise periodicity structure estimates in Shirshov’s height theorem

The Gelfand-Kirillov dimension of l-generated general matrices is equal to (l ? 1)n ² + 1. Due to the Amitzur-Levitsky theorem, the minimal degree of the identity of this algebra is 2n. That is why the essential height of A being an l-generated PI-algebra of degree n over every set of words is greater than (l ? 1)n ²/4 + 1. In this paper we prove that if A has a finite Gelfand-Kirillov dimension, then the number of lexicographically comparable subwords with the period (n ? 1) in each monoid of A is not greater than (l ? 2)(n ? 1). The case of subwords with the period 2 can be generalized to the proof of Shirshov’s height theorem.     

Sperner systems consisting of pairs of complementary subsets

It was proved by Erdös, Ko, and Radó (Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser.12 (1961), 313–320.) that if $\text{A}$ = {;A₁,…, A_l}; consists of k-subsets of a set with n > 2k elements such that A_i ∩ A_j ≠ ? for all i, j then l ? (_k?1^n?1). Schönheim proved that if A₁, …, A_l are subsets of a set S with n elements such that A_i ? A_j, A_i ∩ A_j ≠ ø and A_i ∪ A_j ≠ S for all i ≠ j then $\text{l ? (}{}_{\mathrm{<mtext>[</mtext><mtext>n</mtext><mtext>2</mtext><mtext>]\; ?\; 1</mtext>}}{}^{\mathrm{n\; ?\; 1}}\text{)}$. In this note we prove a common strengthening of these results.     

The variety of the asymptotic values of a real polynomial etale map

A polynomial map F: R² → R² is said to satisfy the Jacobian condition if ∀(X, Y)ϵ R², J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R² → R² that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N), where X → 0, Y is fixed and K, N > 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N) = A(X, Y)ϵ R[X, Y], which ‘capture’ the whole set of asymptotic values of F.     

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates \(f^n\) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).     

An extremal problem among subsets of a set

We determine the maximum size of a family of subsets in {1, 2,…, n} with the property that if A₁, A₂, A₃,… are any members of the family with $\text{∩A}{}_{\mathrm{i}}\text{= ?}$, then ∪A_i = {1, 2,…, n}.     

Interpolation and Parallel Lines

Here we consider 3 interpolation problems for homogeneous polynomials in n n + 1 variables (i.e. for zero-dimensional subschemes Z of Pⁿ) in which the scheme Z is contained in a “ small number ” of “ parallel lines ”; here a finite union D₁ ∪ … ∪ D _x ? Pⁿ of lines is called a set of parallel lines if there is P ∈ Pⁿ such that P ∈ D _i for all i.     

Even permutations as a product of two elements of order five

Let A_n denote the alternating group on n symbols. If n = 5, 6, 7, 10, 11, 12, 13 or n ⩾ 15, every permutation in A_n is the product of two elements of order 5 in A_n. The same is true for n ⩽ 14, except for thirteen types of permutations, namely 3¹, 2², 2⁴, 3³, 2¹3¹4¹, 2²5¹, 2⁵4¹, 1¹, 1², 1³, 1⁴, 3¹1¹, 2⁴1¹. (For example, the permutation (12)(34)(56)(78)(9) is not the product of two elements of order 5 in A₉.)     

Sumsets contained in infinite sets of integers

If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. Erdös conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ? A. This conjecture is still unproved. This paper contains several results on sumsets contained in finite sets of integers. For example, if A is a set of integers of positive upper density, then for any n there exist sets B and F such that B has positive upper density, F has cardinality n, and B + F ? A.     

Retarded equations on the sphere induced by linear equations

Using a Poincaré compactification, the linear homogeneous system of delay equations {x = Ax(t ? 1) (A is an n × n real matrix) induces a delay system π(A) on the sphere Sⁿ. The points at infinity belong to an invariant submanifold S^{n ? 1} of Sⁿ. For an open and dense set of 2 × 2 matrices A with distinct eigenvalues, the system π(A) has only hyperbolic critical points (including the critical points at infinity). For an open and dense set of 2 × 2matrices $\text{A}$ with complex eigenvalues, the nonwandering set at infinity is the union of an odd number of hyperbolic periodic orbits; if $\text{(}\text{det}\text{A}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{<}\text{3π}\text{2}$, the restriction of $\text{π(}\text{A}\text{)}$ to S¹ is Morse-Smale. For n = 1 there exist periodic orbits of period 4 provided that $\text{?A >}\text{π}\text{2}$ and Hopf bifurcation of a center occurs for ?A near $\text{(}\text{π}\text{2}\text{) + 2kπ, k ? Z}$.     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.