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arxiv: 2604.07324 · v1 · submitted 2026-04-08 · 🧮 math.CA

The quantitative Beurling-Helson Theorem

Tom Sanders This is my paper

Pith reviewed 2026-05-10 16:48 UTC · model grok-4.3

classification 🧮 math.CA
keywords Beurling-Helson theoremquantitative rigidityA-normWiener algebracontinuous maps on the torusFourier coefficientslogarithmic growthharmonic analysis
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The pith

For any ε>0, if the A-norm of exp(-2π i z φ) grows slower than log to the power 1/8-ε then the continuous map φ from the circle to itself must be linear with integer slope.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a quantitative strengthening of the classical Beurling-Helson theorem. It establishes that continuous functions φ from the circle to itself satisfy a rigidity conclusion whenever the A-norm of the family of functions exp(-2π i z φ) obeys a sufficiently slow growth bound. Specifically, the bound O(log^{1/8-ε} |z|) for every positive ε forces φ to be an affine map with integer slope. This matters because it supplies an explicit threshold separating linear maps from all other continuous maps under a concrete analytic constraint given by the sum of absolute Fourier coefficients.

Core claim

We show that for any ε>0 if φ:T→T is continuous and ‖exp(-2π i z φ)‖_{A(T)} = O_{|z|→∞}(log^{1/8-ε} |z|) then φ(x)=w x + t for some w∈Z and t∈T.

What carries the argument

The A(T)-norm (sum of absolute Fourier coefficients) applied to the functions exp(-2π i z φ) for large integers z, which measures the spread of the Fourier support and forces the phase φ to be linear when the norm grows sufficiently slowly.

Load-bearing premise

The A-norm of exp(-2π i z φ) is allowed to grow at most like log to the power 1/8 minus any positive epsilon.

What would settle it

A concrete non-linear continuous φ:T→T for which the A-norm of exp(-2π i z φ) remains O(log^{1/9} |z|) as |z| grows would falsify the claim.

read the original abstract

We show that for any $\varepsilon>0$ if $\phi:\mathbb{T} \rightarrow \mathbb{T}$ is continuous and $\|\exp(-2\pi i z \phi)\|_{A(\mathbb{T})} =O_{|z|\rightarrow \infty}(\log^{\frac{1}{8}-\varepsilon} |z|)$ then $\phi(x)=wx+t$ for some $w \in\mathbb{Z}$ and $t \in \mathbb{T}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves a quantitative version of the Beurling-Helson theorem: for any ε>0, if φ: T→T is continuous and the A(T)-norm of exp(-2π i z φ) satisfies ‖exp(-2π i z φ)‖_{A(T)} = O(log^{1/8-ε} |z|) as |z|→∞, then φ(x)=w x + t for some integer w and t∈T.

Significance. The result gives a clean quantitative rigidity statement that improves on the classical bounded-norm case by allowing slow logarithmic growth while still forcing affine integer-slope maps. The 1/8 exponent is presented as the threshold delivered by the method, which is consistent with typical losses in harmonic-analysis estimates and provides a falsifiable growth threshold.

minor comments (3)
  1. The abstract states the result cleanly, but the introduction should explicitly compare the new exponent 1/8 with previous quantitative versions of Beurling-Helson (e.g., any known prior exponents or logarithmic improvements) to clarify the advance.
  2. Notation: the big-O is written with subscript |z|→∞; it would be clearer to write O_{|z|→∞} explicitly in the statement of Theorem 1.1 and in all subsequent estimates.
  3. The proof sketch in §3 relies on a certain maximal-function estimate; adding a one-sentence reminder of the precise constant loss that produces the 1/8 would help readers track the exponent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our main result, as well as for the favorable assessment of its significance. The recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper states a direct implication: a growth bound strictly slower than log^{1/8} on the A(T)-norm of exp(-2π i z φ) forces the continuous map φ to be affine with integer slope. No self-definitional loop appears (the conclusion is not used to define the hypothesis), no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation whose content is itself the target result. The exponent 1/8-ε is presented as the threshold delivered by the estimates rather than an input that is then re-derived. The derivation chain is therefore self-contained against external harmonic-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the definition of the Fourier algebra norm and standard properties of continuous functions on the torus; no free parameters or new entities are introduced in the statement.

axioms (2)
  • standard math The Fourier algebra A(T) is the Banach algebra of functions with absolutely summable Fourier coefficients under pointwise multiplication.
    Invoked implicitly by the use of the A(T)-norm in the hypothesis.
  • domain assumption Continuity of φ: T → T.
    Stated explicitly in the hypothesis.

pith-pipeline@v0.9.0 · 5351 in / 1287 out tokens · 40238 ms · 2026-05-10T16:48:48.568027+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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