Theorems · Theorem · number theory

threeAPFree_frontier

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : TopologicalSpace E] [inst_4 : AddCommMonoid E] [inst_5 : Module 𝕜 E] {s : Set E},
  IsClosed s → StrictConvex 𝕜 s → ThreeAPFree (frontier s)

The frontier of a closed strictly convex set only contains trivial arithmetic progressions. The idea is that an arithmetic progression is contained on a line and the frontier of a strictly convex set does not contain lines.

Defined in

Mathlib.Combinatorics.Additive.AP.Three.Behrend

Cited by

1 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingTopologicalSpaceAddCommMonoidModule