The Ubiquity of Sidon Sets That Are Not I0I_0

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$

A dichotomy property for locally compact groups

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<\omega}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<\omega}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group $G$, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.Comment: To appear in J. of Functional Analysi

Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets

We present a proof of Rider's unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon. Most likely this proof is essentially the one announced by Rider and communicated in a letter to the author around 1979 (lost by him since then). The key fact is a spectral gap property with respect to certain representations of the unitary groups $U(n)$ that holds uniformly over $n$. The proof crucially uses Weyl's character formulae. We survey the results that we obtained 30 years ago using Rider's unpublished results. Using a recent different approach valid for certain orthonormal systems of matrix valued functions, we give a new proof of the spectral gap property that is required to show that the union of two Sidon sets is Sidon. The latter proof yields a rather good quantitative estimate. Several related results are discussed with possible applications to random matrix theory.Comment: v2: minor corrections, v3 more minor corrections v4) minor corrections, last section removed to be included in another paper in preparation with E. Breuillard v5) more minor corrections + two references added. The paper will appear in a volume dedicated to the memory of V. P. Havi

On uniformly bounded orthonormal Sidon systems

In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any uniformly bounded $\psi_2 (C)$-orthonormal system ($\psi_2 (C)$ is a variant of subGaussian)is 2-fold tensor Sidon. This sharpens their result that it is 5-fold tensor Sidon. The proof is somewhat reminiscent of the author's original one for (Abelian) group characters, based on ideas due to Drury and Rider. However, we use Talagrand's majorizing measure theorem in place of Fernique's metric entropy lower bound. We also show that a uniformly bounded orthonormal system is randomly Sidon iff it is 4-fold tensor Sidon, or equivalently $k$-fold tensor Sidon for some (or all) $k\ge 4$. Various generalizations are presented, including the case of random matrices, for systems analogous to the Peter-Weyl decomposition for compact non-Abelian groups. In the latter setting we also include a new proof of Rider's unpublished result that randomly Sidon sets are Sidon, which implies that the union of two Sidon sets is Sidon.Comment: v3: randomly Sidon implies four-fold tensor Sidon. v6: preceding is extended to matrix valued case, also an illustrative-hopefully illuminating-example is presented. Terminolgy is improve

On the Structure of Sets of Large Doubling

We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.Comment: 23 pages, changed title, revised version reflects work of Meyer that we were previously unaware o