Algebraic numbers and Fourier analysis: Salem's third problem
Pith reviewed 2026-05-10 03:06 UTC · model grok-4.3
The pith
If each Fourier transform of a Bernoulli convolution does not vanish at infinity, then their product does.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For algebraic parameters defining a sequence of Bernoulli convolutions, if the Fourier transform of each convolution does not tend to zero at infinity, then the infinite product of these transforms tends to zero at infinity.
What carries the argument
Lang's general formulation of Roth's theorem applied via Weil heights to the algebraic parameters of the Bernoulli convolutions, which controls the possible sizes of the Fourier transforms at large frequencies and forces the product to decay.
If this is right
- The infinite product of the Fourier transforms must tend to zero at infinity.
- The result applies to any finite or countably infinite collection of such Bernoulli convolutions with algebraic parameters.
- The vanishing holds uniformly for the product measure formed by the convolutions.
- The decay of the product is guaranteed by the same diophantine bounds that control individual transforms.
Where Pith is reading between the lines
- Similar vanishing statements might hold for other families of measures whose parameters admit effective diophantine control.
- The technique could be tested numerically for low-degree algebraic numbers to observe the rate at which the product decays.
- Progress on this problem may suggest new approaches to the remaining open questions in Salem's list concerning absolute continuity.
Load-bearing premise
The parameters defining the Bernoulli convolutions are algebraic numbers to which Lang's formulation of Roth's theorem applies directly without additional restrictions on degree or height.
What would settle it
An explicit collection of algebraic numbers λ_j where each associated Fourier transform stays bounded away from zero along some sequence of frequencies going to infinity, yet the infinite product of the transforms does not tend to zero.
read the original abstract
In 1963, Rapha\"el Salem concluded his highly influential book ``Algebraic Numbers and Fourier Analysis'' with a list of four unsolved problems. The first two problems remain wide open while the last problem on the absolute continuity of Bernoulli convolutions has seen significant progress over the years including recent results by Shmerkin and Varj\'u. In this paper, we solve the third problem concerning the vanishing at infinity of the product of Fourier transforms of Bernoulli convolutions each of which does not vanish at infinity. Our solution uses tools in diophantine approximation such as the theory of Weil heights and Lang's general formulation of Roth's theorem.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to solve Salem's third problem (1963) on the vanishing at infinity of the infinite product of Fourier transforms of Bernoulli convolutions, where each individual transform does not vanish at infinity. The solution applies the theory of Weil heights and Lang's formulation of Roth's theorem to the algebraic parameters defining the convolutions, yielding diophantine control over the decay of the product.
Significance. If correct, the result resolves a long-standing open problem at the interface of algebraic number theory and harmonic analysis, complementing progress on Salem's fourth problem by Shmerkin and Varjú. A strength is the reliance on independently established, parameter-free diophantine theorems (Weil heights and Lang-Roth) rather than ad-hoc or data-fitted constructions.
major comments (2)
- [proof of the main theorem (application of Lang-Roth)] The central step applying Lang-Roth to the infinite product: the diophantine estimates must produce uniform decay overcoming the accumulation of factors in ∏ |cos(2π ξ λ^k)| as |ξ|→∞. However, the algebraic numbers arising in the linear forms for partial products up to k∼log|ξ| may have degrees or heights growing with k (or with the specific linear combination from the product), causing the Roth constants to deteriorate and potentially preventing the required vanishing. This needs explicit verification that degrees/heights remain bounded independently of ξ.
- [§2 (setup of Bernoulli convolutions and Fourier transforms)] The reduction from the product of Fourier transforms to control of individual cosine terms assumes that the parameters λ are algebraic of fixed degree to which Lang's formulation applies directly. If the effective degree in the simultaneous approximations for the product grows with the number of convolutions or frequency, the argument requires additional justification or a modified effective version of Roth.
minor comments (2)
- [Abstract] The abstract states that the first two Salem problems remain open; a one-sentence reminder of their statements would improve context for readers unfamiliar with the 1963 list.
- [Introduction] Notation for the infinite product and the indexing of the Bernoulli convolutions should be introduced with a displayed equation early in the paper to avoid ambiguity when referring to 'the product'.
Simulated Author's Rebuttal
We thank the referee for their valuable feedback on our paper solving Salem's third problem. We address each major comment below with clarifications and indicate where revisions will be made to strengthen the exposition.
read point-by-point responses
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Referee: [proof of the main theorem (application of Lang-Roth)] The central step applying Lang-Roth to the infinite product: the diophantine estimates must produce uniform decay overcoming the accumulation of factors in ∏ |cos(2π ξ λ^k)| as |ξ|→∞. However, the algebraic numbers arising in the linear forms for partial products up to k∼log|ξ| may have degrees or heights growing with k (or with the specific linear combination from the product), causing the Roth constants to deteriorate and potentially preventing the required vanishing. This needs explicit verification that degrees/heights remain bounded independently of ξ.
Authors: We thank the referee for highlighting this crucial point regarding uniformity. In the manuscript, the Bernoulli convolution parameter λ is a fixed algebraic number of degree d. The terms λ^k appearing in the Fourier transform product all belong to the same number field of degree d, independent of k and ξ. Thus, the degrees are bounded uniformly. The Weil heights h(λ^k) = k h(λ) grow linearly with k ≈ log|ξ|, but Lang's version of Roth's theorem accounts for this growth in the diophantine bound. Our estimates demonstrate that the product of the cosine terms provides a decay rate that is exponential in log|ξ|, which suffices to overcome the polynomial or logarithmic dependence on the height in the Roth constant. To address the request for explicit verification, we will insert a dedicated paragraph or short lemma after the statement of the main theorem, computing the height growth explicitly and confirming the uniform vanishing. revision: yes
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Referee: [§2 (setup of Bernoulli convolutions and Fourier transforms)] The reduction from the product of Fourier transforms to control of individual cosine terms assumes that the parameters λ are algebraic of fixed degree to which Lang's formulation applies directly. If the effective degree in the simultaneous approximations for the product grows with the number of convolutions or frequency, the argument requires additional justification or a modified effective version of Roth.
Authors: The setup in Section 2 explicitly takes λ to be algebraic of a fixed degree, and the infinite product is formed from powers λ^k within this fixed extension. The linear forms in the diophantine approximations do not involve simultaneous approximations across different fields or increasing degrees; each cosine term corresponds to an element in the fixed field Q(λ). The number of factors up to k ~ log|ξ| does not affect the degree. We will add a clarifying sentence in §2 to emphasize that the effective degree remains constant throughout the product, thereby justifying the direct application of Lang-Roth without modification. revision: yes
Circularity Check
No circularity: external diophantine theorems applied to independent problem
full rationale
The paper's solution to Salem's third problem invokes the theory of Weil heights and Lang's formulation of Roth's theorem to bound the decay of products of Fourier transforms for Bernoulli convolutions parameterized by algebraic numbers. These are classic, independently established results in diophantine approximation, not defined in terms of the target vanishing-at-infinity property, not fitted to the same data, and not justified by self-citation chains. The abstract and reader's summary show no self-definitional steps, no renaming of known results as new derivations, and no load-bearing reliance on prior work by the same author. The derivation remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Lang's general formulation of Roth's theorem on diophantine approximation of algebraic numbers
- standard math Properties of Weil heights on algebraic numbers
Reference graph
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