[Paper Review] Jacques Peyrière et les produits de Riesz

Jean‐Pierre Kahane|arXiv (Cornell University)|Mar 24, 2010

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper by Jean-Pierre Kahane provides a comprehensive retrospective on Jacques Peyrière's foundational work on Riesz products, focusing on their role in harmonic analysis, fractal geometry, and the theory of Sidon sets. It analyzes the mutual singularity and absolute continuity of Riesz product measures under conditions on the coefficients, establishes dimension bounds via ergodic averages, and highlights the pioneering nature of Peyrière's methods in constructing quasi-independent sets with extreme properties in abelian groups.

ABSTRACT

Jacques Peyrière investigated Riesz products associated with a given set of frequencies and the corresponding coefficients : mutual singularity or absolute continuity of the measures defined by two such products, Hausdorff dimensions of the sets carrying such measures, multifractal analysis. The present expository paper starts from Peyrière's results and give some new ways to investigate the same questions. Since Riesz products are also an essential tool for Sidon sets, the characterisations of Sidon sets by Pisier and Bourgain, using quasi-independent sets, are given and commented. The paper ends with an elementary construction of a <> quasi-independent set. The paper is in French. An introduction, in English, explains why it will not be published in the book to which it was intended to contribute.

Motivation & Objective

• To survey and analyze Jacques Peyrière's influential contributions to the theory of Riesz products across harmonic analysis, measure theory, and fractal geometry.
• To establish conditions under which Riesz product measures are mutually singular or absolutely continuous, based on coefficient differences.
• To derive dimension bounds for sets of positive measure under Riesz products using ergodic averages and logarithmic integrals.
• To highlight the foundational role of Peyrière's 1973 and 1975 works in the development of quasi-independent sets and their applications in Sidon set theory.
• To emphasize the methodological innovation and long-term impact of Peyrière's approach in constructing sets with extreme analytic and geometric properties.

Proposed method

• Analyzes Riesz products of the form $ \prod_{j=0}^\infty (1 + \operatorname{Re}(a_j e^{i\lambda_j t})) $ with $ \lambda_j $ satisfying $ \lambda_{j+1}/\lambda_j \geq 3 $, ensuring frequency separation.
• Represents partial products as exponential sums over $ \varepsilon_j \in \{-1,0,1\} $, with distinct frequencies due to the gap condition, leading to a Fourier series representation of the measure $ \mu_a $.
• Applies Wiener's criterion to show that $ \mu_a $ is a diffuse measure under $ |a_j| \leq 1 $.
• Uses orthogonality of functions $ e^{i\lambda_j t} - \frac{1}{2} \overline{a}_j $ in $ L^2(\mu_a) $ to prove singularity when $ \sum |a_j - b_j|^2 = \infty $.
• Derives dimension bounds for $ \mu_a $-positive Borel sets via the limsup and liminf of normalized logarithmic integrals of partial products.
• Constructs quasi-independent sets in abelian groups using Riesz product techniques, showing that such sets can intersect balls in $ \gg k \log k $ points under certain conditions.

Experimental results

Research questions

• RQ1 Under what conditions on the coefficients $ a_j $ and $ b_j $ are two Riesz product measures $ \mu_a $ and $ \mu_b $ mutually singular?
• RQ2 When are two Riesz product measures $ \mu_a $ and $ \mu_b $ mutually absolutely continuous, particularly when $ \sum |a_j - b_j|^2 / (1 - |a_j|) < \infty $?
• RQ3 What are the Hausdorff dimension bounds for Borel sets $ E $ with $ \mu_a(E) > 0 $, in terms of the logarithmic integral of the partial products?
• RQ4 Can quasi-independent sets with $ \gg k \log k $ points in a $ k $-ball be constructed in abelian groups, and under what group conditions?
• RQ5 How do Peyrière's early works (1973, 1975) lay the groundwork for the modern theory of quasi-independent sets and Sidon sets?

Key findings

• If $ \sum |a_j - b_j|^2 = \infty $, then $ \mu_a \perp \mu_b $, meaning the measures are mutually singular.
• If $ \sum |a_j - b_j|^2 / (1 - |a_j|) < \infty $, then $ \mu_a \sim \mu_b $, meaning the measures are mutually absolutely continuous.
• The Hausdorff dimension of any Borel set $ E $ with $ \mu_a(E) > 0 $ satisfies the bounds $ 1 - \limsup_{n \to \infty} \frac{1}{\log \lambda_n} \int \log P_{a,n}(t) \, d\mu_a(t) \leq \dim E \leq 1 - \liminf_{n \to \infty} \frac{1}{\log \lambda_n} \int \log P_{a,n}(t) \, d\mu_a(t) $.
• For $ \lambda_j = 2^j $, the Riesz product measure $ \mu_a $ is well-defined under $ \sup |a_j| < 1 $, and the singularity result holds under the same condition on both $ a_j $ and $ b_j $.
• A quasi-independent set $ \Lambda $ can be constructed such that $ |\Lambda \cap M| \gg k \log_2 k $ for some $ k $-ball $ M $, provided the group contains elements of arbitrarily large order.
• The construction fails in groups like $ \prod_{j=1}^\infty (\mathbb{Z}/p_j\mathbb{Z}) $ with bounded $ p_j $, where intersections are at most $ O(k) $, as shown by Pisier's non-constructive result.