Rajchman Measure in Harmonic Analysis

• Rajchman measure is a finite Borel measure whose Fourier–Stieltjes transform vanishes at infinity, underlining its role in characterizing decay in harmonic analysis.
• It is central in applications such as unique trigonometric series, fractal measures, equidistribution, and Diophantine approximation, linking analytic properties with geometric structure.
• Ongoing research explores algebraic, dynamic, and probabilistic criteria, particularly in self-similar and random walk contexts, to better understand its spectral and ergodic implications.

A Rajchman measure is a finite Borel measure on a locally compact abelian group (or more generally, on a well-defined harmonic-analytic domain) whose Fourier(-Stieltjes) transform vanishes at infinity. Such measures have played a central role in harmonic analysis, uniqueness theory for trigonometric series, and the analytic paper of fractal measures, with applications in equidistribution, spectral theory, and Diophantine approximation. The algebraic and probabilistic conditions under which a measure is Rajchman are the subject of ongoing research, especially concerning self-similar and self-affine measures, random walk stationary measures, and spectral measures for Schrödinger operators.

1. Definition and Basic Properties

Let $G$ be a locally compact abelian group with Pontryagin dual $\widehat{G}$. A finite Borel measure $\mu$ on $G$ is Rajchman if its Fourier–Stieltjes transform

$$\widehat{\mu}(\gamma) = \int_G \overline{\gamma(x)} \, d\mu(x), \quad \gamma\in\widehat{G}$$

satisfies $\widehat{\mu}(\gamma)\to 0$ as $|\gamma|\to\infty$. On $\mathbb{R}$ or $\mathbb{T}$, this means

$$\widehat{\mu}(t) = \int_{\mathbb{R}} e^{-itx}\, d\mu(x) \to 0 \quad \text{as } |t|\to\infty$$

or, on the circle,

$$c_n(\mu) = \int_{\mathbb{T}} z^n\, d\mu(z) \to 0 \quad \text{as } |n|\to\infty$$

A strong characterization is that every absolutely continuous finite measure is Rajchman, and every Rajchman measure is continuous (no pure points), but not all singular continuous measures are Rajchman (Kubrusly, 23 Aug 2025).

The algebra of all Rajchman measures, denoted $M_0(G)$, forms a closed translation-invariant ideal in the measure algebra $M(G)$, and is "solid" under absolute continuity, i.e., an $L$-space (Ghandehari, 2018). On discrete groups, $M_0(G) = \{0\}$, whereas for compact $G$, $M_0(G)$ is typically large (Ghandehari, 2018).

2. Classical and Modern Criteria for the Rajchman Property

Smoothness and Integrability Criteria

For absolutely continuous measures, the Riemann-Lebesgue lemma ensures the Rajchman property. For more general measures, integrability or smoothness of derivatives or transforms can guarantee vanishing of the Fourier transform. For example, if $\mu$ has $L^2$ density, then $|\widehat{\mu}(t)|\leq\|f\|_2\,\sqrt{2\pi}$ and thus $\widehat{\mu}(t)\to0$ as $|t|\to\infty$ (Yakubovich, 2011).

Infinite convolutions and Cantor–type constructions yield singular continuous Rajchman measures—a foundational method in constructing "thin" nontrivial examples (Yakubovich, 2011).

Algebraic and Dynamical Criteria

If the contraction ratios in a self-similar iterated function system are arithmetically incommensurable (i.e., $\log r_i / \log r_j \notin \mathbb{Q}$), or the associated law is non-lattice, then all self-similar measures are Rajchman (Li et al., 2019, Brémont, 2019). Conversely, in commensurable or Pisot-type cases, nondecay can occur.

For Furstenberg stationary measures associated to random walks on $SL_2(\mathbb{R})$, sufficiently high moment conditions (finite second moment) guarantee Rajchman decay of Fourier coefficients (Dinh et al., 2021).

3. Rajchman Measures in Self-Similar and Fractal Geometry

Real Line and Pisot Phenomena

In the setting of Bernoulli convolutions $\nu_\lambda$, the measure is Rajchman except when $1/\lambda$ is a Pisot number (Brémont, 2019). More generally, Rajchman fails only for self-similar measures arising from IFSs in Pisot form; otherwise, vanishing at infinity is generic.

The following table (schematic) captures this dichotomy:

IFS Condition	Rajchman?	Source
Nonlattice, non-Pisot ratios	Yes (always)	(Brémont, 2019)
Pisot form (after change of variables)	Only if absolutely continuous	(Brémont, 2019)
Self-affine with proximal, irreducible group of contractions	Yes	(Li et al., 2019)

Higher Dimensions

In higher dimensions ($\mathbb{R}^d$), self-similar and self-affine measures are Rajchman under natural irreducibility and proximality conditions on the linear part of the IFS (Li et al., 2019, Rapaport, 2021). This includes applications to rectangular multiplicity for trigonometric series and nontrivial Fourier dimension estimates (Li et al., 2019).

Gibbs Measures and Markov Maps

Gibbs measures for the Gauss map are Rajchman when their dimension exceeds $1/2$ (e.g., for the Minkowski question mark function), with explicit power-law decay rates that yield geometric and arithmetic information (Salem sets, equidistribution) (Jordan et al., 2013).

4. Applications and Impact

Uniqueness and Multiplicity in Trigonometric Series

A central application is Menshov–Salem–Zygmund theory, where the presence of a Rajchman measure on a closed set $F$ implies that $F$ is a set of multiplicity: trigonometric series with vanishing coefficients outside $F$ are not unique (Li et al., 2019).

Pointwise Normality and Diophantine Approximation

If $\mu$ is a Rajchman measure, then $\mu$-almost every point is absolutely normal (normal to every base), with no decay rate required—sharpening the Davenport–Erdős–LeVeque criterion for normal number prevalence (Algom, 25 Apr 2025). Quantitative balance between Fourier decay and denominator growth in Diophantine problems yields new Jarník-type theorems and Hausdorff dimension results on fractal sets (Pavlenkov et al., 16 Jan 2024).

Spectral Theory

In operator algebras and quantum Hamiltonians, spectral measures that are Rajchman under mild commutator regularity translate into propagation and scattering results, even when limiting absorption principles are unavailable (Golenia et al., 2017).

Probabilistic and Random Walk Stationary Measures

For measures stationary under random matrix products (e.g., on $SL_2(\mathbb{R})$), a second moment condition suffices for the Rajchman property; renewal-theoretic and spectral techniques provide quantitative decay (Dinh et al., 2021).

Harmonic Analysis of Gaussian Multiplicative Chaos

Random singular measures such as the Gaussian multiplicative chaos on the circle are almost surely Rajchman, with the proof relying on higher-order convolutions and Riemann–Lebesgue-type arguments (Garban et al., 2023).

5. Rajchman Algebras and Banach Algebraic Structure

The algebra $M_0(G)$ of Rajchman measures is a closed ideal in the measure algebra $M(G)$. For non-compact, non-discrete $G$, the maximal ideal space of $M_0(G)$ is rich, containing analytic discs, and the algebra admits nonzero point derivations (Ghandehari, 2018).

For general locally compact (possibly non-abelian) groups $G$, the Rajchman algebra $B_0(G) = B(G)\cap C_0(G)$ (with $B(G)$ the Fourier–Stieltjes algebra) is amenable if and only if $G$ is compact and almost abelian; otherwise, $B_0(G)$ lacks a bounded approximate identity and operator amenability fails (Ghandehari, 2011).

6. Notable Examples and Historic Questions

The Minkowski Question Mark Function and Salem’s Problem

The Minkowski question mark function, a continuous, strictly increasing, singular function, yields a singular measure whose Fourier–Stieltjes coefficients vanish at infinity—affirmatively solving Salem's 1943 problem (Yakubovich, 2011). This is established by an infinite product expansion of the transform, controlled by functional equations, and divergence of the sum $\sum_k(1-|\Phi(2^{-k}t)|)$ for suitable $\Phi$ (Yakubovich, 2011, Yakubovich, 2011, Jordan et al., 2013). Higher derivatives of the transform also vanish at infinity.

7. Open Questions and Ongoing Directions

• Classification of all self-similar and self-affine measures that are Rajchman in higher dimensions, including the impact of algebraic constraints (Pisot phenomena) (Rapaport, 2021).
• Effective equidistribution results quantifying convergence rates in the normality problems for Rajchman measures (Algom, 25 Apr 2025).
• Deeper analytic and cohomological properties of Rajchman algebras; structure of the spectrum, existence of higher-dimensional analytic varieties (Ghandehari, 2018).
• Interaction between operator-theoretic stability concepts (weak stability, weak quasistability) and the Rajchman property for spectral measures (Kubrusly, 23 Aug 2025).

Rajchman measures thus form a foundational class in harmonic analysis, bridging singularity/regularity, geometric measure theory, operator algebras, and ergodic theory. The decay of their Fourier transform stands as a decisive marker of both analytic and arithmetic structure in diverse mathematical objects.