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arxiv: 2602.16482 · v2 · submitted 2026-02-18 · 🧮 math.NT · math.CA· math.CO

Remarks on the inverse Littlewood conjecture

Thomas F. Bloom , Ben Green This is my paper

Pith reviewed 2026-05-15 21:25 UTC · model grok-4.3

classification 🧮 math.NT math.CAmath.CO
keywords Littlewood conjectureFourier analysisadditive combinatoricsarithmetic progressionssumset doublinginverse theorems
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The pith

A bound of K log N on the Fourier L1 norm implies the existence of a large low-doubling subset.

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

The Littlewood conjecture establishes that the Fourier transform of the indicator function of any N-element subset A of the integers has L1 norm at least c log N for some positive constant c. This paper studies the sets that nearly attain this lower bound, specifically those for which the norm is at most K log N. Under this hypothesis the authors extract a subset A prime of A whose size is at least N to the power 0.99 and whose sumset has cardinality bounded by a constant depending only on K times the size of A prime. The resulting additive structure immediately yields that A itself contains arithmetic progressions of any fixed length k once N is large enough in terms of k and K. The argument also produces a modest sharpening of the constant c appearing in the original conjecture.

Core claim

When the L1 norm of the Fourier transform of the indicator of A is at most K log N, there exists a subset A' subset A with |A'| at least N to the power 0.99 such that the sumset A' + A' has size at most K to the power O(1) times |A'|. Consequently, for any k at least 3 and N sufficiently large depending on k and K, the set A contains a k-term arithmetic progression.

What carries the argument

The extraction of a large low-doubling subset from the hypothesis that the L1 norm of the Fourier transform is at most K log N.

If this is right

  • Any such set A contains arithmetic progressions of any fixed length k when N is large enough depending on k and K.
  • The best known lower bound constant c in the Littlewood conjecture receives a slight improvement.
  • Sets whose Fourier L1 norm nearly achieves the minimal possible value must contain large subsets that are additively structured.
  • The inverse Littlewood problem is reduced to controlling the doubling constant of large subsets.

Where Pith is reading between the lines

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

  • Strengthening the size exponent from 0.99 to 1 minus a small epsilon would give quantitative control over how close the whole set is to being an arithmetic progression.
  • The same structural conclusion may apply to other inverse problems that bound the L1 norm of a Fourier transform on the integers.
  • Numerical checks on arithmetic progressions themselves or on random subsets could calibrate the dependence of the doubling constant on K.

Load-bearing premise

That a Fourier L1 norm bounded by K log N is enough to guarantee a large subset whose sumset is controlled by a constant depending only on K.

What would settle it

An explicit construction of sets A with |A| = N large, Fourier L1 norm at most K log N, yet every subset of size N to the 0.99 has sumset size growing with N rather than bounded by a function of K alone.

read the original abstract

The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.

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 investigates the inverse Littlewood conjecture: if A is a finite subset of Z with |A|=N and ||widehat{1_A}||_1 ≤ K log N, then A contains a subset A' with |A'| ≥ N^{0.99} such that |A'+A'| ≪ K^{O(1)} |A'|. As a consequence, for any fixed k≥3 and N large enough (depending on k,K), A contains a k-term arithmetic progression. The paper also derives a modest improvement to the constant c in the classical Littlewood lower bound.

Significance. If the main structural result holds, it supplies a quantitative link between near-extremal L1 Fourier norm and small-doubling subsets, yielding arithmetic-progression conclusions via standard additive-combinatorial arguments. The slight improvement to c is a concrete, if incremental, advance. The work is consistent with the expected landscape of inverse Littlewood-type theorems and contains no visible internal inconsistencies or circular reductions.

minor comments (3)
  1. [Introduction] §1 (Introduction): the Fourier transform notation hat{1_A} is used without an explicit definition or reference to the normalization; a one-line reminder would improve readability for readers outside additive combinatorics.
  2. [Theorem 1.1] Theorem 1.1: the exponent 0.99 in |A'| ≥ N^{0.99} appears as a convenient choice rather than optimal; a brief remark on whether the argument permits any improvement (e.g., to N^{1-ε(K)}) would clarify the result's sharpness.
  3. [Section 4] The proof of the improved constant c is only sketched; a short appendix or paragraph detailing the numerical gain would make the byproduct claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive recommendation for minor revision. The referee's summary accurately captures the main results of the paper, including the structural theorem on large low-doubling subsets and the arithmetic progression consequence, as well as the modest improvement to the Littlewood constant.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives structural consequences (large low-doubling subset A' with |A'| ≥ N^{0.99} and |A'+A'| ≪ K^{O(1)} |A'|, hence long APs) directly from the hypothesis ||1_A hat||_1 ≤ K log N using standard inverse Littlewood-type arguments and additive combinatorics. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the improved constant c is a byproduct of the same analysis rather than an input. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no free parameters, new entities, or ad-hoc axioms are mentioned. The result rests on the known Littlewood theorem plus standard facts from additive combinatorics.

axioms (2)
  • standard math Fourier analysis on the integers and the definition of the L1 norm of the transform
    Used to state the Littlewood constant and the inverse assumption.
  • domain assumption Sets with small doubling contain long arithmetic progressions
    Invoked to pass from the small-sumset conclusion to the AP conclusion.

pith-pipeline@v0.9.0 · 5495 in / 1429 out tokens · 59449 ms · 2026-05-15T21:25:02.600122+00:00 · methodology

discussion (0)

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Reference graph

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