The purpose of this note is to construct an example of a discrete non-abelian group G and a subset E of G, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally “special” Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$-degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.

A subset $E$ of a discrete abelian group is called $\varepsilon$-Kronecker if all $E$-functions of modulus one can be approximated to within ϵ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $\varepsilon$-Kronecker sets with $\varepsilon \,<\,2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.

Let ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ be an increasing sequence of positive integers, and let $f\left( x \right)$ be a real function satisfying

1

$$f\left( x+1 \right)=f\left( x \right),\int\limits_{0}^{1}{f\left( x \right)}dx=0,\text{Va}{{\text{r}}_{\left[ 0,1 \right]}}f<\infty $$

If ${{\lim }_{k\to \infty }}\frac{{{n}_{k+1}}}{{{n}_{k}}}=\infty $ the distribution of

2

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{n}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. In the case

$$1<\underset{k\to \infty }{\mathop{\lim \inf }}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}},\,\,\underset{k\to \infty }{\mathop{\text{lim}\,\text{sup}}}\,\,\frac{{{n}_{k+1}}}{{{n}_{k}}}<\infty$$

there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$, and the limit distribution of (2).

In this paper we prove that any sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying $\lim {{\sup }_{k\to \infty }}{{n}_{k+1}}/{{n}_{k}}=1$ contains a nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ such that for any function satisfying (1) the distribution of

$$\frac{\sum\nolimits_{k=1}^{N}{f\left( {{m}_{k}}x \right)}}{\sqrt{N}}$$

converges to a Gaussian distribution. This result is best possible: for any $\varepsilon >0$ there exists a sequence ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ satisfying lim $\underset{k\to \infty }{\mathop{\sup }}\,\frac{{{n}_{k+1}}}{{{n}_{k}}}\le 1+\varepsilon$ such that for every nontrivial subsequence ${{\left( {{m}_{k}} \right)}_{k\ge 1}}$ of ${{\left( {{n}_{k}} \right)}_{k\ge 1}}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$.

Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

In this note we investigate lacunarity or ‘thin’ subsets in the dual object of a compact group via different classes of summing operators between Banach spaces. In particular, we give characterisations of Sidon and ∧ (p) sets, 2 < p < ∞

Generalized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

A well-known result of Zygmund states that if f ∈ L (log+L) ½ on the circle group T and E is a Hadamard set of integers, then . In this paper we investigate similar results for the classes on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual Γ. These results are obtained as special cases of more general results concerning a new class of lacunary sets Sαβ, 0 < α ≤ β, where a subset E of Γ is an Sα β set if . We also prove partial results on the distinctness of the Sαβ sets in the index β.

In this paper it is shown that the maximal spectral type of a general rank one transformation is given by a kind of generalized Riesz product, with possibly some discrete measure.