Existing in Mathlib · Combinatorics

Roth’s Theorem

Among the first N natural numbers, a set that avoids every nontrivial three-term arithmetic progression must occupy a proportion that tends to zero as N grows.

three-term arithmetic progressions
asymptotic density
Roth numbers
corners theorem
triangle removal

Exact theorem

Exact Mathlib statement

theorem rothNumberNat_isLittleO_id : IsLittleO atTop (fun N ↦ (rothNumberNat N : ℝ)) (fun N ↦ (N : ℝ))

The theorem at a glance

Roth’s Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory additive-combinatorics poster states rothNumberNat N equals little-o of N and traces dense sets through corners and triangle removal to an equally spaced triple. — The largest three-term-progression-free subset of {0,…,N−1} has density tending to zero, with no explicit rate claimed by the selected declaration.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A dense set of lattice points yields a highlighted right-angle corner, which maps to three collinear output points with two equal gaps. — For sufficiently large intervals, a subset of fixed positive density cannot avoid a nontrivial three-term arithmetic progression.

Why it matters

A mathematical landmark

Roth’s theorem is a foundational density result in additive combinatorics. Mathlib’s retained route connects one-dimensional three-term progressions to two-dimensional corners, triangle removal, and Szemerédi regularity.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib’s qualitative asymptotic declaration for the natural-number Roth function. Here rothNumberNat N is the maximum cardinality of a ThreeAPFree subset of Finset.range N = {0,…,N−1}, and ThreeAPFree excludes nontrivial a,b,c in the set with a+c=b+b. The exact conclusion is an IsLittleO atTop relation after coercion to ℝ. It does not state an explicit decay rate, a finite threshold, a modern quantitative bound, or a theorem about longer progressions.

Proof Atlas did not originate Roth’s theorem or Mathlib’s declaration.
The selected declaration gives no explicit rate or numerical threshold.
The selected endpoint concerns three-term arithmetic progressions, not arbitrary progression lengths.
The generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: rothNumberNat_isLittleO_id in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.roth-theorem.v001
Source: Open the pinned upstream reference