Kronecker Sets Research Articles

Isometries with dense windings of the torus in C(M)

Let C(M) be the space of all continuous functions on M⊂ ℂ. We consider the multiplication operator T: C(M) → C(M) defined by Tf(z) = zf(z) and the torus $$ O(M) = \left\{ {f:M \to \mathbb{C} \ntrianglelefteq \left\| f \right\| = \left\| {\frac{1} {f}} \right\| = 1} \right\} $$ . If M is a Kronecker set, then the T-orbits of the points of the torus ½O(M) are dense in ½O(M) and are ½-dense in the unit ball of C(M).

On T-sequences and characterized subgroups

Let X be a compact metrizable abelian group and u = { u n } be a sequence in its dual group X ∧ . Set s u ( X ) = { x : ( u n , x ) → 1 } and T 0 H = { ( z n ) ∈ T ∞ : z n → 1 } . Let G be a subgroup of X. We prove that G = s u ( X ) for some u iff it can be represented as some dually closed subgroup G u of Cl X G × T 0 H . In particular, s u ( X ) is polishable. Let u = { u n } be a T-sequence. Denote by ( X ˆ , u ) the group X ∧ equipped with the finest group topology in which u n → 0 . It is proved that ( X ˆ , u ) ∧ = G u and n ( X ˆ , u ) = s u ( X ) ⊥ . We also prove that the group generated by a Kronecker set cannot be characterized.

Baire category and zero sets

Quasi-all continuous functions have a zero set which is a perfect Kronecker set with Hausdorff dimension zero. To cite this article: T. Körner, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Characterizations of groups generated by Kronecker sets

Ces dernieres annees, depuis l'article [B-D-S], nous avons etudie la possibilite de carateriser les sous-groupes denombrables du tore T = R/Z par des sous-ensembles de Z. Nous considerons ici de nouveaux types de sous-groupes: soit K ⊆ T un ensemble de Kronecker (un ensemble compact sur lequel toute fonction continue f: K → T peut etre approchee uniformement par des caracteres de T) et G le groupe engendre par K. Nous prouvons (theoreme 1) que G peut etre caracterise par un sous-ensemble de Z 2 (au lieu d'un sous-ensemble de Z). Si K est fini, le theoreme 1 implique notre resultat anterieur de [B-S]. Nous montrons egalement (theoreme 2) que si K est denombrable alors G ne peut pas etre caracterise par un sous-ensemble de Z (ou une suite d'entiers) au sens de [B-D-S].

$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, I: arithmetic properties of $\varepsilon$-Kronecker sets

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier–Stieltjes transform of a discrete measure on the dual group.We show that if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.

$\varepsilon$-Kronecker and $I_{0}$ sets in abelian groups, II: sparseness of products of $\varepsilon$-Kronecker sets

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.

ε-Kronecker and I0sets in abelian groups, IV: interpolation by non-negative measures

A subset $E$ of a discrete abelian group is a &#8220;Fatou&#8211;Zygmund interpolation set&#8221; ($F\kern-.75pt ZI_0$ set) if every bounded Hermitian function on $E$ is the restriction of the Fourier&#8211;Stieltjes transform of a discrete, non-negative

On polynomially bounded operators acting on a Banach space

By the Von Neumann inequality every contraction on a Hilbert space is polynomially bounded. A simple example shows that this result does not extend to Banach space contractions. In this paper, we give general conditions under which an arbitrary Banach space contraction is polynomially bounded. These conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers. To do this we use techniques from harmonic analysis, in particular, results concerning thin sets such as Helson sets, Kronecker sets and sets that satisfy spectral synthesis.

Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals of A+.

Article Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals of A+. was published on January 1, 1994 in the journal Journal für die reine und angewandte Mathematik (volume 1994, issue 450).

Bohr compactification and continuous measures

Let G be an LCA group with dual Γ \Gamma . As a consequence of our main result, it is shown that every continuous regular measure μ \mu concentrated on a Kronecker set and with norm &gt; 1 {\text {norm}} &gt; 1 has the property that { | μ ^ | &gt; 1 } \{ |\hat \mu | &gt; 1\} is dense in the Bohr compactification of Γ \Gamma .

On the finite sum of Kronecker sets

On the finite sum of Kronecker sets

On the sum of two Kronecker sets

On the sum of two Kronecker sets

Sets of multiplicity and differentiable functions. II

The paper contains two theorems relating the fine structure of differentiable functions, in one or more dimensions, to the behavior of Fourier-Stieltjes transforms on sets that are small in various ways. In this paper we prove two theorems on the transformation of certain sets, defined as follows. A set E in a metric space is an L-set if there are sequences Ek?-)>O and 6k?_0, and for each k a decomposition E= UI Ei, wherein diam(Ei) k (isi'). For each compact L-set E of real numbers there is a function h of class C'(oo, oo) with h'>O, so that h(E) is a Kronecker set ([2], [3]). The first theorem is a complement to this. THEOREM I. Let co be a monotone, positive function on (0, oc), and c(O +) = O; let C' be the set offunctions q in C1 with p'> O, I p'(a)-p'(b) I _ w)(la-bl)for all real a and b. Then there is a compact L-set E so that g(E) is an MO-set for each q9 in C ,. To prove the theorem we choose a sequence of positive numbers (cj) so that co= 1, wo(cn)<n-2, and c,,+<n -3c0. We now construct finite sets Fn and En; the peculiar construction of Fn is the main point in the argument. Fn is a sequence of n2 elements x(m) = x(O) + MCn + m2cnn-5!2, 1< m _ n2. Here x(O)= -cn-cnn-512 so that x(1)=O. En is then a union of translates of Fn, say Ui (En+aj). Then ao=O, while aj+i-aj=n2c,+n-1/2cn. In different terms, the final term in each translate becomes x(O) in its successor to the right. The number of translates is to be [c,,1lc1n -1316] for n_1. (a) In Fn we have the inequalities Cn < x(m + 1) x(m) < (n2 +? n312)cn < 2n2cn. Thus En has diameter <2n-1/6cn-l. The vector sum E= n En iS then an L-set. (It is somewhat easier to verify that, for large enough r, the subset Received by the editors January 15, 1971. AMS 1969 subject classifications. Primary 4272, 4615.

Topics on Kronecker sets

Topics on Kronecker sets

Kronecker Sets and Metric Properties of M 0 -Sets

Kronecker Sets and Metric Properties of M 0 -Sets

The functions which operate on Fourier-Stieltjes transforms of L-subalgebras of [formula omitted

An L-subalgebra OL of the algebra M T of Borel measures on the circle group T is a closed subalgebra OL ⊆ M( T ) such that μ ∈ OL implies L 1(μ)⊆ OL . A subalgebra OL is symmetric if μ∈ OL implies μ ∈ OL , where μ(−E) = μ(−E) , ( E any Borel set). A function F : [−1, 1] → C operates on a set OL of Fourier-Stieltjes transforms if μ ∈ OL , μ ( Z )⊆[−1,1] imply F ∘ μ is a Fourier-Stieltjes transform. We have these results: If F : [−1, 1] → C operates on the Fourier-Stieltjes transforms of an infinite dimensional symmetric L-subalgebra OL , then F is analytic in a neighborhood of 0. If OL contains a measure concentrated on a Dirichlet set, then F is analytic in a neighborhood of [−1, 1]. If OL contains a continuous measure concentrated on a Kronecker set, then F is entire. Related results are given.

Kronecker sets and metric properties of $M\sb{0}$-sets

A method for constructing both sets of multiplicity and Kronecker sets within a given set of multiplicity is derived from the work of Ivashev-Musatov; it is shown that the Hausdorff measures and other measures are essentially distinct. Finally, an improvement of a theorem of Salem is obtained, using PyateckilShapiro's theorem on non-M sets. 1. The class of complex Borel measures , on the real axis, such that fi(u)-0 as Iul->oo, is denoted R, and a closed set is called Mo if it supports some measure ,u#O of class R. With a measure , the entire space L1(1) is contained in R [7, I, p. 143], so that we mostly study probability measures in R. The most striking examples of non-MO sets are the Kronecker sets: A compact linear set E is a K-set if each continuous function on E of modulus 1 admits uniform approximation on E by characters Z(x)= e2Ur (-oo O, then E1r(E2+x) is MO for an x-set of positive Lebesgue measure. A sharp converse is true. THEOREM 1. Let F be a closed set of Lebesgue measure in(F)=O. Then there exists an MO-set E so that (F+x) r)E is a K-set for every real x. Theorem 1 is a consequence of a more general assertion; for closed sets F and numbers r>O let m(F, r)=m{x: dist(x, F) g(r)for all r<rO(F). Theorem 1' is derived from a very general theorem in which Lebesgue measure plays no special role. THEOREM 2. Let It be a probability of class R and of closedl support S; let (Tk)Y be a sequence of closed sets w ith lim y (T)=0. Then there exists an Received by the editors January 19, 1972. AMS (MOS) subject classJifcations (1969). Primary 4250, 4252; Secondary 4256.

Some results on Kronecker, Dirichlet and Helson sets

We construct the following: a perfect non Dirichlet set every proper closed subset of which is Kronecker, A weak Kronecker set which is not an R set; an independent countable Dirichlet set which is not Kronecker; a collection of q-disjoint Kronecker sets whose union is independent but Helson 1/q; A countable collection of disjoint Kronecker sets whose union is closed and independent but not Helson: a perfect independent Dirichlet set which is not Helson.

A problem on Kronecker sets

A problem on Kronecker sets

Spectral synthesis for the Kronecker sets

Spectral synthesis for the Kronecker sets