M. Laczkovich

Spectral synthesis on discrete Abelian groups

We prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

Harmonic analysis on discrete Abelian groups

Let G be an Abelian group and let C G denote the linear space of all complex-valued functions defined on G equipped with the product topology. We prove that the following are equivalent. (i) Every nonzero translation invariant closed subspace of C G contains an exponential; that is, a nonzero multiplicative function. (ii) The torsion free rank of G is less than the continuum.

THE BOREL STRUCTURE OF ITERATES OF CONTINUOUS-FUNCTIONS

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f ∈C[0,1] mapping [0,1] into itself. If f ∈C, f k will denote the kth iterate of f and we put C k = { f k : f ∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I , we let C( f ) denote the set of points at which f is locally constant, i.e. We let N denote the set of positive integers and N N denote the Baire space of sequences of positive integers.

The number of homothetic subsets

We investigate the maximal number S(P, n) of subsets of a set of n elements homothetic to a fixed set P. Elekes and Erdős proved that S(P, n) > cn 2 if | P | = 3 or the elements of P are algebraic. For | P | ≥ 4 we show that S(P, n) > cn 2 if and only if every quadruple in P has an algebraic cross ratio. Moreover, there is a sequence S n of numbers such that \(S(P,n) \asymp S_{n}\) whenever | P | = 4 and the cross ratio of P is transcendental.

Tilings of polygons with similar triangles

We prove that if a polygonP is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices ofP. IfP is a rectangle then, apart from four “sporadic” cases, the triangles of the decomposition must be right triangles. Three of these “sporadic” triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angleα such that tanα is a totally positive algebraic number. Most of the proofs are based on the following general theorem: if a convex polygonP is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices ofP belong to the field generated by the cotangents of the angles of the triangles in the decomposition.

Convexity conditions and intersections with smooth functions

A continuous function that agrees with each member of a family Sof smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when Sconsists of the family of polynomials of degree at most n, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function f agrees with each polynomial of degree at most n in only a finite set, then f is ( n + 1)-convex or ( n + 1)-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.

Tilings of the square with similar rectangels

LetR(u) denote the rectangle of sidesu and 1. We prove that the square can be decomposed into finitely many rectangles similar toR(u) if and only ifu is algebraic and each of its conjugates lies in the open half-plane Re(z)>0.

Analytic subgroups of the reals

We prove that every analytic proper subgroup of the reals can be covered by an F-sigma null set. We also construct a proper Borel subgroup G of the reals that cannot be covered by countably many sets A(i) such that A(i) + A(i) is nowhere dense for every i.

PARADOXICAL DECOMPOSITIONS USING LIPSCHITZ FUNCTIONS

§1. Introduction and main results. A map f: A → R (A xs2282 R) is called piecewise contractive if there is a finite partition A = A1xs222A … xs222A An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).

Measurable Darboux functions

We investigate how certain Darboux-like properties of real functions (including connectivity, almost continuity, and peripheral continuity) are related to each other within certain measurability classes (including the classes of Lebesgue measureable, Borel, and Baire-1 functions).