Almost Affine Invariance Over Prime Fields: Green Problem 90

Jie Ma, Max Wenqiang Xu, Quanyu Tang

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Pith reviewed 2026-05-14 18:01 UTC · model grok-4.3

classification 🧮 math.CO math.DSmath.GRmath.NT

keywords affine invarianceprime fieldssymmetric differencedensity 1/2Green's problemfinite fieldsadditive combinatoricsthresholds

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The pith

A density-1/2 set in the prime field F_p is almost affine invariant under all maps ax+b with |a| and |b| at most o(log p).

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

The paper shows that for any subset A of the finite field with p elements having density one half, the symmetric difference between A and its image under an affine map ax+b is o(p) for all maps with coefficient bounds |a| and |b| no larger than K if and only if K grows slower than log p. This establishes the sharp threshold for simultaneous almost affine invariance and resolves an open problem posed by Ben Green. A reader should care because it clarifies the largest family of affine symmetries that large subsets can preserve up to negligible error. The result implies that logarithmic growth marks the point where such invariance must fail for some map.

Core claim

Let A be a subset of F_p with density 1/2. Then A is almost affine invariant under every affine transformation φ(x) = ax + b with |a|, |b| ≤ K and a ≠ 0 simultaneously if and only if K = o(log p).

What carries the argument

Almost affine invariance defined via |A Δ φ(A)| = o(p) for affine maps φ, with the threshold K on the size of coefficients a and b.

If this is right

The o(p) symmetric difference bound holds uniformly for all affine maps with |a|, |b| = o(log p).
For any K = ω(log p), there exist density-1/2 sets A that fail to be almost invariant under at least one such map.
The threshold is sharp, giving the exact scale at which simultaneous invariance breaks.
This holds without additional assumptions on A beyond the density being 1/2.

Where Pith is reading between the lines

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

If the threshold is sharp, then logarithmic scales likely control other near-invariance properties in finite fields.
Analogous thresholds might exist for invariance under higher degree maps or different group actions.
Computational checks for small p could verify the transition around log p.
The result suggests exploring whether density 1/2 is necessary or if it holds for other densities.

Load-bearing premise

The o(p) bound on the symmetric difference is uniform over all qualifying affine maps rather than varying with the specific coefficients or depending on further properties of A.

What would settle it

A construction of a density-1/2 set A together with coefficients a and b where |a| or |b| exceeds any multiple of log p but |A Δ (aA + b)| remains Ω(p) would confirm the threshold; failure to find such a counterexample for K = (log p)^2 would falsify the claimed limit.

read the original abstract

Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $\phi(x)=ax+b$ if \[|A \triangle \phi(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $\phi(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90.

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.

Circularity Check

0 steps flagged

No circularity: direct determination of threshold via independent argument

full rationale

The abstract states a direct determination that the threshold for simultaneous almost affine invariance is K=o(log p), solving an external open problem. No equations, definitions, or steps are shown that reduce the claimed threshold to a fitted parameter, self-definition, or self-citation chain. The o(p) error is asserted uniformly without visible construction from the input density-1/2 assumption in a tautological way. This is the common case of a self-contained proof against an external benchmark (Green's problem), yielding no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of affine maps over prime fields and the combinatorial notion of symmetric difference being o(p); no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Standard algebraic properties of the prime field F_p
The paper works inside the usual finite-field setting with no additional assumptions stated.

pith-pipeline@v0.9.0 · 5394 in / 1144 out tokens · 61990 ms · 2026-05-14T18:01:22.393496+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages

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