### A class of ${I}_{0}$-sets

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For some time it has been known that there exist continuous Helson curves in ${\mathbf{R}}^{2}$. This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson $k$-manifold in ${\mathbf{R}}^{nk}$ for $n\ge k+1$. We present a construction of a Helson 2-manifold in ${\mathbf{R}}^{3}$. With modification, our method should even suffice to prove that there are Helson hypersurfaces in any ${\mathbf{R}}^{n}$.

Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define ${r}_{N}^{\left(m\right)}\left(\omega \right)$ to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that ${r}_{N}^{\left(m\right)}\left(\omega \right)<K$ for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

In [3] I showed that there are Helson sets on the circle 𝕋 which are not of synthesis, by constructing a Helson set which was not of uniqueness and so automatically not of synthesis. In [2] Kaufman gave a substantially simpler construction of such a set; his construction is now standard. It is natural to ask whether there exist Helson sets which are of uniqueness but not of synthesis; this has circulated as an open question. The answer is "yes" and was also given in [3, pp. 87-92] but seems to...

Answering a question of Pisier, posed in [10], we construct an L-set which is not a finite union of translates of free sets.

It is a long standing open problem whether Sidon subsets of ℤ can be dense in the Bohr compactification of ℤ ([LR]). Yitzhak Katznelson came closest to resolving the issue with a random process in which almost all sets were Sidon and and almost all sets failed to be dense in the Bohr compactification [K]. This note, which does not resolve this open problem, supplies additional evidence that the problem is delicate: it is proved here that if one has a Sidon set which clusters at even one member of...

It is known that the dual of a compact, connected, non-abelian group may contain no infinite central Sidon sets, but always does contain infinite central $p$-Sidon sets for $p\>1.$ We prove, by an essentially constructive method, that the latter assertion is also true for every infinite subset of the dual. In addition, we investigate the relationship between weighted central Sidonicity for a compact Lie group and Sidonicity for its torus.

Pisier's characterization of Sidon sets as containing proportional-sized quasi-independent subsets is given a sharper form for groups with only a finite number of elements having orders a power of 2. No such improvement is possible for a general Sidon subset of a group having an infinite number of elements of order 2. The method used also gives several sharper forms of Ramsey's characterization of Sidon sets as containing proportional-sized I₀-subsets in a uniform way, again in groups containing...

In this paper, we introduce and study the notion of completely bounded ${\Lambda}_{p}$ sets (${\Lambda}_{p}^{cb}$ for short) for compact, non-abelian groups G. We characterize ${\Lambda}_{p}^{cb}$ sets in terms of completely bounded ${L}^{p}\left(G\right)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are ${\Lambda}_{p}$ sets for all p < ∞, but are not ${\Lambda}_{p}^{cb}$ for any p ≥ 4. This is done by showing that the space of completely bounded ${L}^{p}\left(G\right)$ multipliers...

We study the Complex Unconditional Metric Approximation Property for translation invariant spaces ${C}_{\Lambda}\left(\right)$ of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which ${C}_{\Lambda}\left(\right)$ has (ℂ-UMAP); though these sets are such that ${L}_{\Lambda}^{\infty}\left(\right)$ contains functions which are not continuous, we show that there is a linear invariant lifting from these ${L}_{\Lambda}^{\infty}\left(\right)$ spaces into the Baire class 1 functions.

On montre que si G est un groupe abélien localment compact non diskret à base dénombrable d'ouverts, alors la famille des fermés de synthèse pour l'algèbre de Fourier A(G) est une partie coanalytique non borélienne de ℱ(G), l'ensemble des fermés de G muni de la structure borélienne d'Effros. On généralise ainsi un résultat connu dans le cas du groupe 𝕋.