Riesz Products: Theory and Applications

• Riesz products are probability measures generated as weak-* limits of multiplicative families of nonnegative trigonometric polynomials with lacunary frequencies and bounded coefficients.
• Their construction relies on lacunarity conditions and Fourier expansion techniques that ensure nonnegativity, normalization, and convergence, enabling analysis of measure singularity and absolute continuity.
• Applications span harmonic analysis, Sidon sets, and ergodic theory, with extensions to higher-dimensional settings like spheres and Bohr compactifications, offering insights into multifractal spectra and spectral gaps.

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 (Kahane, 2010, Doubtsov, 2024, Doubtsov, 2024).

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 $T=\mathbb{R}/2\pi\mathbb{Z}$0-th partial product is

$T=\mathbb{R}/2\pi\mathbb{Z}$1

Each factor is nonnegative and has average $T=\mathbb{R}/2\pi\mathbb{Z}$2, so $T=\mathbb{R}/2\pi\mathbb{Z}$3 and

$T=\mathbb{R}/2\pi\mathbb{Z}$4

Hence

$T=\mathbb{R}/2\pi\mathbb{Z}$5

is a probability measure (Kahane, 2010).

The lacunarity condition ensures that in the Fourier expansion of $T=\mathbb{R}/2\pi\mathbb{Z}$6, no two distinct multi-indices contribute the same frequency. Equivalently, $T=\mathbb{R}/2\pi\mathbb{Z}$7 is the partial sum of a trigonometric series indexed by $T=\mathbb{R}/2\pi\mathbb{Z}$8, with frequencies $T=\mathbb{R}/2\pi\mathbb{Z}$9. By positivity and uniform boundedness of $\mathbb{R}$0, the sequence $\mathbb{R}$1 is weak-$\mathbb{R}$2 compact, and $\mathbb{R}$3 converges in the weak-$\mathbb{R}$4 sense to a probability measure $\mathbb{R}$5, the Riesz product measure associated with the data $\mathbb{R}$6 (Kahane, 2010).

Closely related normalizations appear elsewhere in the literature. On the unit circle $\mathbb{R}$7, one also writes

$\mathbb{R}$8

for real $\mathbb{R}$9 and $S^{2n-1}\subset\mathbb{C}^n$0, or more generally

$S^{2n-1}\subset\mathbb{C}^n$1

for real frequencies $S^{2n-1}\subset\mathbb{C}^n$2 and complex $S^{2n-1}\subset\mathbb{C}^n$3 with $S^{2n-1}\subset\mathbb{C}^n$4 (Doubtsov, 2024, Neretin, 2018). 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 $S^{2n-1}\subset\mathbb{C}^n$5 and $S^{2n-1}\subset\mathbb{C}^n$6 built on the same frequency sequence $S^{2n-1}\subset\mathbb{C}^n$7 but with different coefficients $S^{2n-1}\subset\mathbb{C}^n$8 and $S^{2n-1}\subset\mathbb{C}^n$9. If

$T=\mathbb{R}/2\pi\mathbb{Z}$0

then $T=\mathbb{R}/2\pi\mathbb{Z}$1. If, in addition,

$T=\mathbb{R}/2\pi\mathbb{Z}$2

then $T=\mathbb{R}/2\pi\mathbb{Z}$3, so the two measures are mutually absolutely continuous (Kahane, 2010).

The singularity argument uses centered functions

$T=\mathbb{R}/2\pi\mathbb{Z}$4

which form an $T=\mathbb{R}/2\pi\mathbb{Z}$5-orthogonal system with norms uniformly bounded away from $T=\mathbb{R}/2\pi\mathbb{Z}$6 and $T=\mathbb{R}/2\pi\mathbb{Z}$7. Choosing a square-summable real sequence $T=\mathbb{R}/2\pi\mathbb{Z}$8 such that $T=\mathbb{R}/2\pi\mathbb{Z}$9, one obtains convergent $dt/2\pi$0-series under $dt/2\pi$1 and $dt/2\pi$2 whose limits are incompatible on any set of positive overlap; this forces disjointness (Kahane, 2010).

Several refinements sharpen this coefficient-based classification. Parreau showed that if $dt/2\pi$3 for all $dt/2\pi$4 and $dt/2\pi$5, then $dt/2\pi$6. Kilmer and Saeki extended absolute-continuity results to random Riesz products (Kahane, 2010).

For the classical circle model, the standard dichotomy is the one associated with Zygmund: if $dt/2\pi$7, 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 $dt/2\pi$8 measure; if $dt/2\pi$9, one obtains a singular continuous measure (Doubtsov, 2024, Doubtsov, 2024). A plausible implication is that lacunarity alone does not determine the measure class; once the frequency system is fixed, the decisive parameter is the $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$0-geometry of the coefficients.

Generalized Riesz products admit analogous calculus. For

$\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$1

under the hypotheses stated by el Abdalaoui and Nadkarni, the finite-block ratios

$\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$2

converge in $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$3 to $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$4, giving a Radon–Nikodym formula internal to the product structure (Abdalaoui et al., 2013).

3. Hausdorff dimension and multifractal structure

For a classical Riesz product $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$5 on $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$6, Peyrière studied not only support properties but also the Hausdorff dimensions carried by the measure. In the constant-ratio case, where $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$7 is constant and $\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$8, define

$\{d_0,d_1,d_2,\dots\}\subset\mathbb{N}$9

Then the limit

$d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$0

exists, and the lower and upper Hausdorff dimensions coincide: $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$1 In the specific case $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$2 and $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$3, one computes

$d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$4

(Kahane, 2010).

Under weaker assumptions, when the sequence is only lacunary in the sense that $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$5 is bounded below and above by fixed constants $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$6 and $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$7, Billingsley’s approach yields two-sided bounds: $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$8 The proof uses Billingsley’s ergodic-theoretic theorem applied to the shift on the symbolic space $d_{j+1}\ge 3(d_0+d_1+\cdots+d_j),$9 corresponding to the Fourier expansion of the product (Kahane, 2010).

Multifractal analysis enters through the local scaling exponent

$\{a_j\}_{j\ge 0}$0

when the limit exists, and the multifractal spectrum

$\{a_j\}_{j\ge 0}$1

Peyrière, and later Fan and Barral for general multiplicative measures, relate this spectrum to the Legendre transform of the $\{a_j\}_{j\ge 0}$2-spectrum

$\{a_j\}_{j\ge 0}$3

Under suitable differentiability hypotheses,

$\{a_j\}_{j\ge 0}$4

This formalism situates Riesz products within the broader theory of multiplicative cascades and nonuniform scaling measures (Kahane, 2010).

4. Sidon sets and quasi-independence

Riesz products are also an essential tool in the theory of Sidon sets. A subset $\{a_j\}_{j\ge 0}$5 is called a Sidon set if the Fourier coefficient norm and the uniform norm are equivalent on trigonometric polynomials with spectrum in $\{a_j\}_{j\ge 0}$6: there exists $\{a_j\}_{j\ge 0}$7 such that for every finitely supported sequence $\{a_j\}_{j\ge 0}$8,

$\{a_j\}_{j\ge 0}$9

A sufficient combinatorial condition is quasi-independence: $|a_j|\le 1$0 is quasi-independent if every finite relation

$|a_j|\le 1$1

is trivial (Kahane, 2010).

In the Pisier–Bourgain characterization cited by Kahane, every finite union of quasi-independent sets is a Sidon set. Conversely, $|a_j|\le 1$2 is Sidon if and only if there exists $|a_j|\le 1$3 such that for every positive measure $|a_j|\le 1$4 supported on $|a_j|\le 1$5, one can find a quasi-independent subset $|a_j|\le 1$6 with $|a_j|\le 1$7 (Kahane, 2010).

The Riesz-product proof of the direct implication constructs

$|a_j|\le 1$8

for suitable signs $|a_j|\le 1$9, where $T=\mathbb{R}/2\pi\mathbb{Z}$00 and each $T=\mathbb{R}/2\pi\mathbb{Z}$01 is quasi-independent. The product is nonnegative, has integral $T=\mathbb{R}/2\pi\mathbb{Z}$02, and can be arranged so that at a point where a trigonometric polynomial attains its supremum, each factor captures a large portion of the $T=\mathbb{R}/2\pi\mathbb{Z}$03-mass. The resulting estimate gives

$T=\mathbb{R}/2\pi\mathbb{Z}$04

for some absolute $T=\mathbb{R}/2\pi\mathbb{Z}$05 (Kahane, 2010).

Kahane’s exposition ends with an explicit construction of a “big” quasi-independent set. Binary matrices $T=\mathbb{R}/2\pi\mathbb{Z}$06 of size $T=\mathbb{R}/2\pi\mathbb{Z}$07 are built inductively, starting from

$T=\mathbb{R}/2\pi\mathbb{Z}$08

and then

$T=\mathbb{R}/2\pi\mathbb{Z}$09

A combinatorial argument preserves quasi-independence of the columns, with

$T=\mathbb{R}/2\pi\mathbb{Z}$10

After embedding these columns at carefully chosen frequencies $T=\mathbb{R}/2\pi\mathbb{Z}$11, one obtains a quasi-independent set $T=\mathbb{R}/2\pi\mathbb{Z}$12 meeting each mesh

$T=\mathbb{R}/2\pi\mathbb{Z}$13

in at least about $T=\mathbb{R}/2\pi\mathbb{Z}$14 points. This shows that the logarithmic factor in the classical upper bound $T=\mathbb{R}/2\pi\mathbb{Z}$15 cannot be replaced by a constant, even for purely quasi-independent sets (Kahane, 2010).

5. Higher-dimensional and generalized settings

Later work recast Riesz products in several nonclassical environments.

Setting	Building blocks	Representative conclusion
Circle $T=\mathbb{R}/2\pi\mathbb{Z}$16	$T=\mathbb{R}/2\pi\mathbb{Z}$17 or $T=\mathbb{R}/2\pi\mathbb{Z}$18	dichotomy, Radon–Nikodym and Mahler formulas
Sphere $T=\mathbb{R}/2\pi\mathbb{Z}$19	$T=\mathbb{R}/2\pi\mathbb{Z}$20	mutual singularity and dimension bounds
$T=\mathbb{R}/2\pi\mathbb{Z}$21 and $T=\mathbb{R}/2\pi\mathbb{Z}$22	$T=\mathbb{R}/2\pi\mathbb{Z}$23, $T=\mathbb{R}/2\pi\mathbb{Z}$24	singularity criteria and spectral realizations

On the unit sphere $T=\mathbb{R}/2\pi\mathbb{Z}$25, Doubtsov uses a Ryll–Wojtaszczyk sequence $T=\mathbb{R}/2\pi\mathbb{Z}$26 with

$T=\mathbb{R}/2\pi\mathbb{Z}$27

Given a lacunary index set $T=\mathbb{R}/2\pi\mathbb{Z}$28 with $T=\mathbb{R}/2\pi\mathbb{Z}$29 and amplitudes $T=\mathbb{R}/2\pi\mathbb{Z}$30, the weak-$T=\mathbb{R}/2\pi\mathbb{Z}$31 limit

$T=\mathbb{R}/2\pi\mathbb{Z}$32

defines a probability measure on $T=\mathbb{R}/2\pi\mathbb{Z}$33. A key structural fact is the slice decomposition: writing $T=\mathbb{R}/2\pi\mathbb{Z}$34 with $T=\mathbb{R}/2\pi\mathbb{Z}$35 and $T=\mathbb{R}/2\pi\mathbb{Z}$36, the measure disintegrates into classical one-dimensional Riesz products on circles (Doubtsov, 2024).

This slicing principle underlies the sphere analog of Peyrière’s theorem. For standard sphere Riesz products, if $T=\mathbb{R}/2\pi\mathbb{Z}$37, then there exists a sequence of unitary operators $T=\mathbb{R}/2\pi\mathbb{Z}$38 such that the rotated triples produce mutually singular measures. As a corollary, if $T=\mathbb{R}/2\pi\mathbb{Z}$39, then $T=\mathbb{R}/2\pi\mathbb{Z}$40, whereas if $T=\mathbb{R}/2\pi\mathbb{Z}$41, then after an appropriate choice of unitary rotations one has $T=\mathbb{R}/2\pi\mathbb{Z}$42. For generalized sphere products built from non-homogeneous holomorphic polynomials with large spectral gaps, the same singularity conclusion holds without rotations: if $T=\mathbb{R}/2\pi\mathbb{Z}$43, then $T=\mathbb{R}/2\pi\mathbb{Z}$44, and in particular $T=\mathbb{R}/2\pi\mathbb{Z}$45 implies $T=\mathbb{R}/2\pi\mathbb{Z}$46 (Doubtsov, 2024).

Doubtsov also estimates energy and Hausdorff dimensions on $T=\mathbb{R}/2\pi\mathbb{Z}$47. For a Riesz triple $T=\mathbb{R}/2\pi\mathbb{Z}$48, define

$T=\mathbb{R}/2\pi\mathbb{Z}$49

Then

$T=\mathbb{R}/2\pi\mathbb{Z}$50

When $T=\mathbb{R}/2\pi\mathbb{Z}$51, one has $T=\mathbb{R}/2\pi\mathbb{Z}$52, hence $T=\mathbb{R}/2\pi\mathbb{Z}$53. The same paper proves that any pluriharmonic measure on $T=\mathbb{R}/2\pi\mathbb{Z}$54 satisfies $T=\mathbb{R}/2\pi\mathbb{Z}$55, and examples show this bound is sharp (Doubtsov, 2024).

Generalized Riesz products on the circle replace the elementary factors by analytic trigonometric polynomials $T=\mathbb{R}/2\pi\mathbb{Z}$56 satisfying $T=\mathbb{R}/2\pi\mathbb{Z}$57-normalization and weak-$T=\mathbb{R}/2\pi\mathbb{Z}$58 convergence of the measures $T=\mathbb{R}/2\pi\mathbb{Z}$59. In this setting, el Abdalaoui and Nadkarni establish a weak dichotomy governed by the product of constant terms $T=\mathbb{R}/2\pi\mathbb{Z}$60: if $T=\mathbb{R}/2\pi\mathbb{Z}$61, the partial analytic products converge weakly to $T=\mathbb{R}/2\pi\mathbb{Z}$62 in $T=\mathbb{R}/2\pi\mathbb{Z}$63; if $T=\mathbb{R}/2\pi\mathbb{Z}$64, they converge in $T=\mathbb{R}/2\pi\mathbb{Z}$65 to a nonzero function in $T=\mathbb{R}/2\pi\mathbb{Z}$66. They also derive the Mahler measure formula

$T=\mathbb{R}/2\pi\mathbb{Z}$67

for the outer constant terms $T=\mathbb{R}/2\pi\mathbb{Z}$68, and the Radon–Nikodym formula discussed earlier (Abdalaoui et al., 2013).

On the Bohr compactification $T=\mathbb{R}/2\pi\mathbb{Z}$69, one considers

$T=\mathbb{R}/2\pi\mathbb{Z}$70

The measures $T=\mathbb{R}/2\pi\mathbb{Z}$71 converge weak-$T=\mathbb{R}/2\pi\mathbb{Z}$72 to a generalized Riesz product $T=\mathbb{R}/2\pi\mathbb{Z}$73. Bourgain’s singularity criterion takes the form

$T=\mathbb{R}/2\pi\mathbb{Z}$74

If an infinite subsequence has rationally independent parameters

$T=\mathbb{R}/2\pi\mathbb{Z}$75

then the limit measure is singular with respect to Haar measure. Kac’s complex central limit theorem yields

$T=\mathbb{R}/2\pi\mathbb{Z}$76

which is the quantitative input forcing vanishing of the multiplicative $T=\mathbb{R}/2\pi\mathbb{Z}$77-averages along an independent subsequence (Abdalaoui, 2012).

6. Spectral, arithmetic, and finite-product applications

Riesz products occur naturally as spectral measures in ergodic theory. For rank-one measure-preserving transformations, Neretin defines

$T=\mathbb{R}/2\pi\mathbb{Z}$78

where the polynomials $T=\mathbb{R}/2\pi\mathbb{Z}$79 are determined by cutting-and-stacking data. A gap property implies convergence of Fourier coefficients and hence weak convergence to a probability measure $T=\mathbb{R}/2\pi\mathbb{Z}$80. The associated Koopman operator is unitarily equivalent to multiplication by $T=\mathbb{R}/2\pi\mathbb{Z}$81 on $T=\mathbb{R}/2\pi\mathbb{Z}$82, so $T=\mathbb{R}/2\pi\mathbb{Z}$83 is exactly the spectral measure of a cyclic vector. Under mild arithmetic growth conditions, $T=\mathbb{R}/2\pi\mathbb{Z}$84 has no atom and no absolutely continuous part, hence is purely singular continuous (Neretin, 2018).

For rank-one flows on $T=\mathbb{R}/2\pi\mathbb{Z}$85, generalized Riesz products are built from

$T=\mathbb{R}/2\pi\mathbb{Z}$86

together with the Fejér kernel

$T=\mathbb{R}/2\pi\mathbb{Z}$87

The weak limit

$T=\mathbb{R}/2\pi\mathbb{Z}$88

is the spectral measure of a natural cyclic vector, and the continuous part of the maximal spectral type is equivalent to $T=\mathbb{R}/2\pi\mathbb{Z}$89. In the same setting, the Salem–Zygmund central limit theorem is extended to $T=\mathbb{R}/2\pi\mathbb{Z}$90, Radon–Nikodym and weak Mahler-measure formulas are proved, and a flow-version of Banach’s problem is solved via $T=\mathbb{R}/2\pi\mathbb{Z}$91-flat polynomials. By contrast, the “exponential staircase” polynomials are shown not to be locally flat on any fixed interval $T=\mathbb{R}/2\pi\mathbb{Z}$92 (Abdalaoui, 2020).

Generalized Riesz products also provide a flexible arithmetic tool for constructing IP-Dirichlet measures on $T=\mathbb{R}/2\pi\mathbb{Z}$93. Given an increasing sequence $T=\mathbb{R}/2\pi\mathbb{Z}$94, one forms

$T=\mathbb{R}/2\pi\mathbb{Z}$95

from nonnegative trigonometric polynomials $T=\mathbb{R}/2\pi\mathbb{Z}$96 of bounded spectrum. Under the dissociation condition

$T=\mathbb{R}/2\pi\mathbb{Z}$97

the measures converge weak-$T=\mathbb{R}/2\pi\mathbb{Z}$98 to $T=\mathbb{R}/2\pi\mathbb{Z}$99, and long gaps in the Fourier support imply continuity. With the explicit choice

$\mathbb{R}$00

one gets lower bounds on Fourier coefficients sufficient to ensure that

$\mathbb{R}$01

In particular, if

$\mathbb{R}$02

then there exists a continuous IP-Dirichlet measure along $\mathbb{R}$03 (Grivaux, 2012).

Finite Riesz products support an additional $\mathbb{R}$04-theory. For

$\mathbb{R}$05

Bonami, Latała, Nayar, and Tkocz show that if the lacunarity ratio $\mathbb{R}$06 is large enough, then linear combinations of the normalized finite products realize an isomorphic embedding of $\mathbb{R}$07 into $\mathbb{R}$08: $\mathbb{R}$09 They give explicit thresholds, including

$\mathbb{R}$10

and corresponding lower-bound thresholds (Bonami et al., 2018).

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 $\mathbb{R}$11 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 (Abdalaoui et al., 2013, Abdalaoui, 2020).