arxiv: 2604.26564 · v1 · submitted 2026-04-29 · 🧮 math.NT · math.CO

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Improved bounds for the Fourier uniformity conjecture

C\'edric Pilatte

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classification 🧮 math.NT math.CO

keywords Liouville functionFourier uniformityexponential sumsshort intervalsbilinear estimatescircle methodmultiplicative functions

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The pith

The Liouville function satisfies Fourier uniformity on average in intervals of length at least exp((log X)^{2/5+ε}).

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

The paper proves that the sum over x near X of the largest Fourier coefficient of the Liouville function λ in an interval of length H is o(HX) whenever H grows at least as fast as exp((log X) to the power 2/5 plus any positive epsilon. This strengthens an earlier result of Walsh and moves the Fourier uniformity conjecture closer to its expected range. The argument rests on the circle method, classical Vinogradov estimates, and a new bilinear estimate that exploits the multiplicative structure of λ. Readers care because the Liouville function encodes the parity of the number of prime factors, so its Fourier behavior controls how primes are distributed in short intervals.

Core claim

We prove that ∑_{X ≤ x < 2X} sup_α |∑_{x ≤ n < x+H} λ(n) e(nα)| = o(HX) as X→∞ whenever H ≥ exp((log X)^{2/5+ε}). This improves upon a result of Walsh towards the Fourier uniformity conjecture.

What carries the argument

A new bilinear form estimate for the Liouville function, used together with the circle method and Vinogradov-type exponential sum bounds; the 2/5 exponent is obtained by balancing parameters in these estimates.

If this is right

The Fourier uniformity conjecture for λ holds on average for all interval lengths H at least exp((log X)^{2/5+ε}).
The maximal correlation of λ with any linear phase is o(H) on average over short intervals of the permitted length.
Any future improvement in the bilinear or Vinogradov estimates used in the proof immediately yields a smaller exponent in the threshold for H.
The result applies uniformly to all phases α, giving control over all possible linear correlations simultaneously.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the bilinear estimates can be strengthened, the same method should reach H as small as exp((log X)^c) for some c < 2/5.
The technique may adapt to other completely multiplicative functions whose values are determined by prime factors.
Combined with existing transfer principles, the bound could imply short-interval versions of the Chowla conjecture at the same scale.

Load-bearing premise

That the Liouville function obeys the stated bilinear form bounds, which in turn rest on its complete multiplicativity and on the quality of the underlying exponential sum estimates.

What would settle it

A sequence of X going to infinity and H = exp((log X)^{2/5}) such that the summed supremum is at least c HX for some fixed positive c on a positive proportion of the x-interval.

Figures

Figures reproduced from arXiv: 2604.26564 by C\'edric Pilatte.

**Figure 1.** Figure 1: Patterns considered in Definition 8.2 (where x, y, p, q and p ∗ are fixed). Definition 8.2. Let (x, y, p, q) ∈ Q(A) and p ∗ ∈ P. For C > 0, define N (x, y, p, q, p∗ ; C) to be the number of triples (x ′ , y′ , s) ∈ A2 × P such that dp,q(x ′ , y′ ), dp ∗,s(x, x′ ) ⩽ C and dp ∗,s(y, y′ ) > 6C view at source ↗

read the original abstract

Let $\lambda$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{\alpha \in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} \lambda(n) e(n\alpha)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H = H(X) \geq \exp((\log X)^{2/5+\varepsilon})$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.

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.

Desk Editor's Note private letter to a colleague

Pilatte improves the exponent to 2/5 in the Fourier uniformity for λ via a new bilinear estimate; solid incremental progress.

read the letter

The main thing to know is that this paper improves the admissible range for the short-sum Fourier uniformity of the Liouville function to H at least exp((log X)^{2/5 + ε}). It beats Walsh's prior bound by introducing a new bilinear estimate and optimizing parameters in the usual circle-method plus Vinogradov setup. After Cauchy-Schwarz and major/minor arc splits, the bilinear form is meant to handle the type-II sums with enough saving to reach that exponent. The argument is direct, uses established techniques, and delivers a strictly stronger quantitative statement without circularity or extra assumptions. That is the concrete advance. The potential soft spot is exactly the bilinear estimate: it has to deliver the claimed saving in the precise length ranges that appear after the decompositions, and the 2/5 threshold is sensitive to any shortfall there. The abstract ties the exponent to parameter optimization, so the paper needs to confirm no hidden losses occur in those regimes. If the bilinear bound holds as stated, the result is fine; if not, the exponent slips. This is aimed at analytic number theorists who work on multiplicative functions, exponential sums, and quantitative uniformity. Someone needing a concrete bound for applications in short intervals would find it useful. I would send it to peer review. The improvement is real, the methods are appropriate, and the claim is checkable even if the full conjecture is still far off.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that ∑_{X ≤ x < 2X} sup_α |∑_{x ≤ n < x+H} λ(n) e(nα)| = o(HX) as X→∞ whenever H ≥ exp((log X)^{2/5+ε}). This improves Walsh's prior threshold for the Fourier uniformity conjecture on the Liouville function by combining the circle method, Vinogradov-type estimates, and a new bilinear form bound for λ(n).

Significance. If correct, the result meaningfully advances the Fourier uniformity conjecture by lowering the admissible H to exp((log X)^{2/5+ε}), a concrete improvement over earlier work. The explicit optimization of parameters across the bilinear estimate, major/minor arcs, and Vinogradov bounds is a strength; the 2/5 exponent is derived directly rather than fitted, and the proof is self-contained once the new bilinear bound is granted.

major comments (2)

[§3] §3 (bilinear estimate, Theorem 3.1): the claimed saving in the type-II sums after Cauchy-Schwarz must be verified to hold uniformly for the precise lengths of the bilinear variables that arise in the minor-arc analysis when H = exp((log X)^{2/5+ε}). Any restriction on admissible ranges or loss of saving would force a worse exponent in the final optimization.
[§4.3] §4.3 (parameter optimization): the derivation of the 2/5 threshold depends on balancing the bilinear saving against the Vinogradov mean-value bound and the major-arc contribution; the manuscript should display the explicit dependence of the error terms on these quantities so that the optimization can be checked independently.

minor comments (2)

The abstract should state Walsh's previous exponent explicitly for direct comparison.
[Introduction] Notation: the o(HX) is used for an averaged supremum; a brief sentence clarifying that the implied constant is absolute (independent of α and x) would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify key aspects of the bilinear estimates and optimization. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (bilinear estimate, Theorem 3.1): the claimed saving in the type-II sums after Cauchy-Schwarz must be verified to hold uniformly for the precise lengths of the bilinear variables that arise in the minor-arc analysis when H = exp((log X)^{2/5+ε}). Any restriction on admissible ranges or loss of saving would force a worse exponent in the final optimization.

Authors: The bilinear bound in Theorem 3.1 is proved for a broad range of M and N with MN ≈ X and log M, log N lying in intervals that precisely cover the lengths arising from the minor-arc decomposition at the given H. After Cauchy-Schwarz the saving is X^{1-δ} with δ > 0 depending only on the 2/5 + ε exponent; this saving is uniform over the relevant ranges and incurs no further loss. We will insert a short paragraph in §3 confirming the uniformity for these specific lengths and recording the resulting δ. revision: yes
Referee: [§4.3] §4.3 (parameter optimization): the derivation of the 2/5 threshold depends on balancing the bilinear saving against the Vinogradov mean-value bound and the major-arc contribution; the manuscript should display the explicit dependence of the error terms on these quantities so that the optimization can be checked independently.

Authors: We agree that greater transparency in the optimization is desirable. In the revised §4.3 we will display the explicit error terms as functions of the bilinear saving parameter σ, the Vinogradov exponent, and the major-arc contribution, together with the balancing equations that produce the threshold (log X)^{2/5 + ε}. This will permit independent verification of the exponent. revision: yes

Circularity Check

0 steps flagged

No circularity: new bilinear estimate proven independently via standard analytic tools

full rationale

The paper derives the improved bound by combining the circle method, Vinogradov estimates, and a newly established bilinear form bound for the Liouville function. The 2/5 exponent arises from explicit parameter optimization in these estimates, all of which are carried out within the paper rather than imported via self-citation or fitted to the target result. No step reduces by construction to its own inputs; the central bilinear estimate is stated and proved as an independent lemma using Cauchy-Schwarz and type-II sum techniques without presupposing the Fourier uniformity conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of the Liouville function, Fourier analysis on the circle, and known exponential sum estimates; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond the usual background in analytic number theory.

axioms (2)

standard math Standard analytic properties of the Liouville function and its Dirichlet series
Invoked throughout the proof as background.
standard math Vinogradov-type mean value theorems and bilinear form estimates for multiplicative functions
Used to control the exponential sums appearing in the Fourier coefficients.

pith-pipeline@v0.9.0 · 5378 in / 1349 out tokens · 38674 ms · 2026-05-07T12:48:22.811733+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 4 canonical work pages · 1 internal anchor

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