--- title: 'Riesz Products: Theory and Applications' url: https://www.emergentmind.com/topics/riesz-products type: topic --- # Riesz Products: Theory and Applications Riesz products are probability measures obtained as weak-\(*\) limits of multiplicative families of nonnegative trigonometric polynomials. In the classical one-dimensional setting on the circle \(T=\mathbb{R}/2\pi\mathbb{Z}\), they are built from lacunary frequencies and bounded coefficients, and they provide canonical examples for studying mutual singularity, absolute continuity, Hausdorff dimension, multifractal spectra, and Sidon phenomena. Subsequent work has extended the construction to generalized products on the circle, the Bohr compactification of \(\mathbb{R}\), the real line, the unit sphere \(S^{2n-1}\subset\mathbb{C}^n\), and pluriharmonic measures on tori, while preserving the central role of dissociation, spectral gaps, and multiplicative structure [1003.4600], [2405.14609], [2404.02652]. ## 1. Classical construction on the circle Let \(T=\mathbb{R}/2\pi\mathbb{Z}\) be equipped with normalized Lebesgue measure \(dt/2\pi\). A classical Riesz product is built from a lacunary sequence of frequencies \(\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}\) satisfying \[ d_{j+1}\ge 3(d_0+d_1+\cdots+d_j), \] and a sequence of complex coefficients \(\{a_j\}_{j\ge 0}\) with \(|a_j|\le 1\). The \(N\)-th partial product is \[ P_N(t)=\prod_{j=0}^{N}\bigl(1+\Re(a_j e^{i d_j t})\bigr). \] Each factor is nonnegative and has average \(1\), so \(P_N(t)\ge 0\) and \[ \int_T P_N(t)\,\frac{dt}{2\pi}=1. \] Hence \[ \mu_N(dt)=P_N(t)\,\frac{dt}{2\pi} \] is a probability measure [1003.4600]. The lacunarity condition ensures that in the Fourier expansion of \(P_N\), no two distinct multi-indices contribute the same frequency. Equivalently, \(P_N\) is the partial sum of a trigonometric series indexed by \(\varepsilon_j\in\{-1,0,1\}\), with frequencies \(\sum_j \varepsilon_j d_j\). By positivity and uniform boundedness of \(\|\mu_N\|=1\), the sequence \(\{\mu_N\}\) is weak-\(*\) compact, and \(\mu_N\) converges in the weak-\(*\) sense to a probability measure \(\mu_a\), the Riesz product measure associated with the data \((d_j,a_j)\) [1003.4600]. Closely related normalizations appear elsewhere in the literature. On the unit circle \(\mathbb{T}\), one also writes \[ \prod_{k=1}^\infty \bigl(1+2a_k\cos(n_k\theta)\bigr) \] for real \(a_k\in(-1,1)\) and \(n_{k+1}/n_k\ge 3\), or more generally \[ P_n(x)=\prod_{k=1}^n \bigl(1+\Re(a_k e^{i\omega_k x})\bigr) \] for real frequencies \(\omega_k\) and complex \(a_k\) with \(|a_k|\le 1\) [2405.14609], [1806.10884]. These variants preserve the basic mechanism: positivity, normalization, and convergence under suitable sparseness or lacunarity assumptions. ## 2. Measure class: singularity, equivalence, and dichotomy A central theorem of Peyrière considers two Riesz product measures \(\mu_a\) and \(\mu_b\) built on the same frequency sequence \(\{d_j\}\) but with different coefficients \(\{a_j\}\) and \(\{b_j\}\). If \[ \sum_{j=0}^{\infty} |a_j-b_j|^2=\infty, \] then \(\mu_a\perp \mu_b\). If, in addition, \[ \sum_{j=0}^{\infty}\frac{|a_j-b_j|^2}{1-|a_j|}<\infty, \] then \(\mu_a\ll \mu_b\ll \mu_a\), so the two measures are mutually absolutely continuous [1003.4600]. The singularity argument uses centered functions \[ \phi_j(t)=\Re(a_j e^{i d_j t})-E_{\mu_a}[\Re(a_j e^{i d_j t})], \] which form an \(L^2(\mu_a)\)-orthogonal system with norms uniformly bounded away from \(0\) and \(\infty\). Choosing a square-summable real sequence \(\{c_j\}\) such that \(\sum c_j(a_j-b_j)=\infty\), one obtains convergent \(L^2\)-series under \(\mu_a\) and \(\mu_b\) whose limits are incompatible on any set of positive overlap; this forces disjointness [1003.4600]. Several refinements sharpen this coefficient-based classification. Parreau showed that if \(|a_j|=|b_j|\) for all \(j\) and \(\sum |a_j-b_j|^2<\infty\), then \(\mu_a\sim \mu_b\). Kilmer and Saeki extended absolute-continuity results to random Riesz products [1003.4600]. For the classical circle model, the standard dichotomy is the one associated with Zygmund: if \(\sum |a_k|^2<\infty\), the Riesz product is absolutely continuous with respect to Lebesgue measure, and in the formulation of Doubtsov’s dimensional paper it is in fact an \(L^2\) measure; if \(\{a_k\}\notin \ell^2\), one obtains a singular continuous measure [2405.14609], [2404.02652]. A plausible implication is that lacunarity alone does not determine the measure class; once the frequency system is fixed, the decisive parameter is the \(\ell^2\)-geometry of the coefficients. Generalized Riesz products admit analogous calculus. For \[ \mu=\prod_{j=1}^\infty |P_j|^2,\qquad \nu=\prod_{j=1}^\infty |Q_j|^2, \] under the hypotheses stated by el Abdalaoui and Nadkarni, the finite-block ratios \[ R_n(z)=\prod_{j=1}^n \frac{|P_j(z)|}{|Q_j(z)|} \] converge in \(L^1(\nu)\) to \(\sqrt{d\mu/d\nu}\), giving a Radon–Nikodym formula internal to the product structure [1307.6513]. ## 3. Hausdorff dimension and multifractal structure For a classical Riesz product \(\mu=\mu_a\) on \(T\), Peyrière studied not only support properties but also the Hausdorff dimensions carried by the measure. In the constant-ratio case, where \(d_{j+1}/d_j=r\ge 3\) is constant and \(|a_j|\le a<1\), define \[ S_N=\int_T \log P_N(t)\,d\mu(t). \] Then the limit \[ L=\lim_{N\to\infty}\frac{S_N}{\log d_N} \] exists, and the lower and upper Hausdorff dimensions coincide: \[ \dim_H(\mu)=1-L. \] In the specific case \(a_j=a>0\) and \(r=4\), one computes \[ \dim_H(\mu)=1-\frac{\frac12\log(1+a^2)}{\log 4} \] [1003.4600]. Under weaker assumptions, when the sequence is only lacunary in the sense that \(d_{j+1}/d_j\) is bounded below and above by fixed constants \(>1\) and \(|a_j|<1\), Billingsley’s approach yields two-sided bounds: \[ 1-\limsup_{N\to\infty}\frac{1}{\log d_N}\int_T \log P_N \le \dim_H^-(\mu)\le \dim_H^+(\mu)\le 1-\liminf_{N\to\infty}\frac{1}{\log d_N}\int_T \log P_N. \] The proof uses Billingsley’s ergodic-theoretic theorem applied to the shift on the symbolic space \(\{-1,0,1\}^{\mathbb{N}}\) corresponding to the Fourier expansion of the product [1003.4600]. Multifractal analysis enters through the local scaling exponent \[ \alpha(t)=\lim_{r\to 0}\frac{\log \mu(B(t,r))}{\log r}, \] when the limit exists, and the multifractal spectrum \[ f(\alpha)=\dim_H\{t:\alpha(t)=\alpha\}. \] Peyrière, and later Fan and Barral for general multiplicative measures, relate this spectrum to the Legendre transform of the \(L^q\)-spectrum \[ \tau(q)=\liminf_{N\to\infty}\frac{-\log \int P_N^q}{\log d_N}. \] Under suitable differentiability hypotheses, \[ f(\alpha)=\inf_{q\in\mathbb{R}} [q\alpha-\tau(q)]. \] This formalism situates Riesz products within the broader theory of multiplicative cascades and nonuniform scaling measures [1003.4600]. ## 4. Sidon sets and quasi-independence Riesz products are also an essential tool in the theory of Sidon sets. A subset \(A\subset\mathbb{Z}\) is called a Sidon set if the Fourier coefficient norm and the uniform norm are equivalent on trigonometric polynomials with spectrum in \(A\): there exists \(S<\infty\) such that for every finitely supported sequence \(\{x_n\}_{n\in A}\), \[ \sum_{n\in A}|x_n| \le S\cdot \sup_t\left|\sum_{n\in A} x_n e^{i n t}\right|. \] A sufficient combinatorial condition is quasi-independence: \(A\) is quasi-independent if every finite relation \[ \sum_j \varepsilon_j n_j=0,\qquad \varepsilon_j\in\{-1,0,1\}, \] is trivial [1003.4600]. In the Pisier–Bourgain characterization cited by Kahane, every finite union of quasi-independent sets is a Sidon set. Conversely, \(A\subset\mathbb{Z}\) is Sidon if and only if there exists \(c(A)>0\) such that for every positive measure \(\omega\) supported on \(A\), one can find a quasi-independent subset \(A'\subset A\) with \(\omega(A')\ge c\,\omega(A)\) [1003.4600]. The Riesz-product proof of the direct implication constructs \[ R(t)=\prod_{j=1}^k \prod_{n\in A_j} \bigl(1+\Re(\varepsilon_{j,n}e^{i n t})\bigr) \] for suitable signs \(\varepsilon_{j,n}\in\{\pm 1\}\), where \(A=A_1\cup\cdots\cup A_k\) and each \(A_j\) is quasi-independent. The product is nonnegative, has integral \(1\), and can be arranged so that at a point where a trigonometric polynomial attains its supremum, each factor captures a large portion of the \(\ell^1\)-mass. The resulting estimate gives \[ S(A)\le C\cdot k\cdot 2^{k-1} \] for some absolute \(C\) [1003.4600]. Kahane’s exposition ends with an explicit construction of a “big” quasi-independent set. Binary matrices \(A_v\) of size \(2^v\times N_v\) are built inductively, starting from \[ A_1= \begin{bmatrix} 1&1&1\\ 1&-1&0 \end{bmatrix}, \] and then \[ A_{v+1}=[\,A_v\ \ A_v\ \ I_v\ \ A_v\ \ -A_v\,]. \] A combinatorial argument preserves quasi-independence of the columns, with \[ N_v=2^{v-1}(v+2). \] After embedding these columns at carefully chosen frequencies \(B_m\), one obtains a quasi-independent set \(A\subset\mathbb{Z}\) meeting each mesh \[ M(k,\{y_j\},E)=\left\{\sum_{j=1}^k n_j y_j:(n_1,\dots,n_k)\in E\right\}, \qquad E=\{-1,0,1\}^k, \] in at least about \(k\log_2 k\) points. This shows that the logarithmic factor in the classical upper bound \(|A\cap M|=O(k\log(1+k))\) cannot be replaced by a constant, even for purely quasi-independent sets [1003.4600]. ## 5. Higher-dimensional and generalized settings Later work recast Riesz products in several nonclassical environments. | Setting | Building blocks | Representative conclusion | |---|---|---| | Circle \(\mathbb{T}\) | \(\prod (1+2a_k\cos(n_k\theta))\) or \(\prod |P_j|^2\) | dichotomy, Radon–Nikodym and Mahler formulas | | Sphere \(S^{2n-1}\) | \(\prod (1+a_kR_{j_k}+\overline{a_k}\,\overline{R_{j_k}})\) | mutual singularity and dimension bounds | | \(\mathrm{Bohr}\,\mathbb{R}\) and \(\mathbb{R}\) | \(|\prod P_k|^2\,dh\), \(|\prod P_n|^2 K_s(t)\,dt\) | singularity criteria and spectral realizations | On the unit sphere \(S^{2n-1}\subset\mathbb{C}^n\), Doubtsov uses a Ryll–Wojtaszczyk sequence \(\{R_j\}\) with \[ R_j\in H(j,0),\qquad \|R_j\|_{L^\infty(S^{2n-1})}=1,\qquad \|R_j\|_{L^2(S^{2n-1})}\ge \delta>0. \] Given a lacunary index set \(J=\{j_k\}\) with \(j_{k+1}/j_k\ge 3\) and amplitudes \(a_k\in\mathbb{D}\), the weak-\(*\) limit \[ \Pi(R,J,a)=\lim_{N\to\infty}\prod_{k=1}^N \bigl(1+a_kR_{j_k}(\zeta)+\overline{a_k}\,\overline{R_{j_k}(\zeta)}\bigr) \] defines a probability measure on \(S^{2n-1}\). A key structural fact is the slice decomposition: writing \(\zeta=\lambda\xi\) with \(\lambda\in\mathbb{T}\) and \(\xi\in\mathbb{CP}^{n-1}\), the measure disintegrates into classical one-dimensional Riesz products on circles [2405.14609]. This slicing principle underlies the sphere analog of Peyrière’s theorem. For standard sphere Riesz products, if \(a-b\notin \ell^2\), then there exists a sequence of unitary operators \(\{U_k\}\subset U(n)\) such that the rotated triples produce mutually singular measures. As a corollary, if \(\sum |a_k|^2<\infty\), then \(\Pi(R,J,a)\ll \sigma\), whereas if \(\sum |a_k|^2=\infty\), then after an appropriate choice of unitary rotations one has \(\Pi(R,J,a)\perp \sigma\). For generalized sphere products built from non-homogeneous holomorphic polynomials with large spectral gaps, the same singularity conclusion holds without rotations: if \(a-b\notin\ell^2\), then \(\Pi(R,J,a)\perp \Pi(R,J,b)\), and in particular \(\sum|a_k|^2=\infty\) implies \(\Pi(R,J,a)\perp \sigma\) [2404.02652]. Doubtsov also estimates energy and Hausdorff dimensions on \(S^{2n-1}\). For a Riesz triple \((R,J,a)\), define \[ \alpha_0= \max\left\{ 0,\; \limsup_{k\to\infty} \frac{(k-1)\log 2+\sum_{\ell=1}^{k-1}\log(1+|a_\ell|)}{\log j_k} \right\}. \] Then \[ \dim_E \Pi \ge 2n-1-\alpha_0, \qquad \dim_H \Pi \ge 2n-1-\alpha_0. \] When \(\sum |a_k|^2<\infty\), one has \(\alpha_0=0\), hence \(\dim_H\Pi=2n-1\). The same paper proves that any pluriharmonic measure on \(\mathbb{T}^n\) satisfies \(\dim_H\mu\ge n-1\), and examples show this bound is sharp [2405.14609]. Generalized Riesz products on the circle replace the elementary factors by analytic trigonometric polynomials \(P_j\) satisfying \(L^2\)-normalization and weak-\(*\) convergence of the measures \(|P_{i_1}\cdots P_{i_k}|^2\,dz\). In this setting, el Abdalaoui and Nadkarni establish a weak dichotomy governed by the product of constant terms \(b=\prod a_0^{(j)}\): if \(b=0\), the partial analytic products converge weakly to \(0\) in \(L^2\); if \(b>0\), they converge in \(L^1\) to a nonzero function in \(H^2\). They also derive the Mahler measure formula \[ M(\mu)=\prod_{j=1}^\infty |\alpha_j|^2 \] for the outer constant terms \(\alpha_j\), and the Radon–Nikodym formula discussed earlier [1307.6513]. On the Bohr compactification \(\mathrm{Bohr}\,\mathbb{R}\), one considers \[ P_k(t)=\frac{1}{\sqrt{p_k}}\sum_{j=0}^{p_k-1} e^{\,i\,t\,(j h_k+s_{k,j})}, \qquad \mu_N=\left|\prod_{k=0}^N P_k(t)\right|^2 dh(t). \] The measures \(\mu_N\) converge weak-\(*\) to a generalized Riesz product \(\mu\). Bourgain’s singularity criterion takes the form \[ \inf_{n_1<\cdots2\sum_{j=1}^k m_j n_j, \] the measures converge weak-\(*\) to \(\sigma\), and long gaps in the Fourier support imply continuity. With the explicit choice \[ P_k(e^{2\pi i t})= \frac{2}{m_k+2}\left| \sum_{j=1}^{m_k+1} \sin\!\left(\frac{j\pi}{m_k+2}\right)e^{2\pi i j t} \right|^2, \] one gets lower bounds on Fourier coefficients sufficient to ensure that \[ \widehat{\sigma}\Bigl(\sum_{k\in F} n_k\Bigr)\to 1 \quad\text{as }\min F\to\infty. \] In particular, if \[ \sum_k \left(\frac{n_k}{n_{k+1}}\right)^2<\infty, \] then there exists a continuous IP-Dirichlet measure along \((n_k)\) [1209.2884]. Finite Riesz products support an additional \(L^p\)-theory. For \[ P_N(t)=\prod_{j=1}^N (1+\cos(n_j t)), \qquad P_N^{(p)}=\frac{P_N}{\|P_N\|_{L^p}}, \] Bonami, Latała, Nayar, and Tkocz show that if the lacunarity ratio \(n_{j+1}/n_j\ge r\) is large enough, then linear combinations of the normalized finite products realize an isomorphic embedding of \(\ell^p\) into \(L^p(T)\): \[ C_1\,\|c\|_{\ell^p} \le \left\|\sum_{j=1}^N c_j P_j^{(p)}\right\|_{L^p(T)} \le C_2\,\|c\|_{\ell^p}. \] They give explicit thresholds, including \[ r\ge d_{\rm up}(p)=80\,p^2,\qquad C_2(p)=(16p)^{p+1}, \] and corresponding lower-bound thresholds [1805.10918]. A recurring simplification is to identify Riesz products only with singular continuous measures on the circle. The record surveyed here is more varied: classical products do yield singular measures outside the \(\ell^2\) regime, but generalized constructions also produce absolutely continuous measures, positive Mahler measure, Lebesgue spectral type, higher-dimensional singular measures on spheres, and arithmetic measures with prescribed Fourier behavior [1307.6513], [2007.03684]. Source: https://www.emergentmind.com/topics/riesz-products