Dirichlet's approximation theorem

Proved

FamousTheorems.exists_norm_nsmul_le

by cm_beta · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

mathlibnumber-theory

Dirichlet's approximation theorem. For any real $\theta$ and any $N$ there is $1 \le n \le N$ with $\lVert n\theta\rVert \le 1/N$ , distance measured to the nearest integer. Equivalently every irrational admits infinitely many rationals with $|\theta - p/q| < 1/q^2$ . The proof is the pigeonhole principle applied to the fractional parts of $\theta, 2\theta, \ldots, N\theta$ , Dirichlet's original use of that principle in 1842. The exponent 2 is essentially optimal: Hurwitz sharpened the constant, and Liouville's theorem shows algebraic numbers cannot be approximated much better, which is where transcendence proofs begin. Formalization note. Stated on AddCircle, where the norm is distance to the nearest integer multiple. The result is Mathlib's AddCircle.exists_norm_nsmul_le.

Preamble

import Mathlib

Formal statement

namespace FamousTheorems

universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25

open Filter Set Topology DirectSum

theorem exists_norm_nsmul_le :
    ∀ {T : ℝ} [hT : Fact (0 < T)] (ξ : AddCircle T) {n : ℕ}, 
    0 < n → ∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ T / ↑(n + 1) := by sorry

end FamousTheorems

Source

Listed in Mathlib's curated theorem manifests; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.

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