Sharp affine-invariance range for half-density subsets of finite fields

How large can K be if a half-density subset of Fₚ is almost invariant under every affine map x↦ax+b with 1≤|a|≤K and |b|≤K? The proposed sharp range is K=o(log p).

Why it matters and what counts as progress

Why it matters: A sharp logarithmic threshold closes the gap between amenability constructions and earlier much weaker upper restrictions.

What counts as progress: Construct almost-invariant sets throughout the range and prove a matching upper restriction, or quantify the invariance error.

How the problem developed

13 May 2026 · Threshold result [1]
Jie Ma, Quanyu Tang, and Max Wenqiang Xu, with ChatGPT 5.4 assistance Resolved Green Problem 90: a half-density set with near invariance under every specified affine map exists exactly when K=o(log p).

AI contribution

Almost Affine Invariance Over Prime Fields: Green Problem 90

Jie Ma, Quanyu Tang, Max Wenqiang Xu

Ma, Tang and Xu prove that half-density subsets of Fₚ can be almost invariant under every affine map with 1≤|a|≤K and |b|≤K exactly when K=o(log p). The AI-attributed new ingredient is the q-adic valuation formulation used for the sharp upper bound.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFull