Results on the irrationality exponent of π and the tools of its proof, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Dirichlet's approximation theorem — Proposition 1.3. Every irrational x has infinitely many coprime p/q with 0 < |x − p/q| < 1/q^2; hence mu(x) >= 2.
• Liouville's theorem — Proposition 1.4. A real algebraic number of degree n >= 2 satisfies |x − p/q| >= c q^(−n) for all p/q; hence mu(x) <= n.
• Formal inverse function theorem — Lemma 1.1. A local homomorphism of power series rings in n variables with invertible linear part is an isomorphism.
• Exponential of commuting derivations — Lemma 1.2. For commuting derivations D_i of a Q-algebra R, f -> sum_alpha D^alpha f b^alpha / alpha! is a ring homomorphism R -> R[[b]].
• Weighted transverse multiplicity — Theorem 3.1. If polynomials of weighted degree <= N have all derivatives of cost < epsilon N along a commuting frame vanishing on a component Z of codimension k, then prod_{a in A} rho_a <= k! epsilon^(−k) prod_{b in B} kappa_b for all coordinate and frame normal bases A, B of Z.
• Separated weights — Lemma 1.2. Thresholds for w_1, ..., w_m, depending only on the fixed data and the earlier weights (not on the centres), make the weights separated.
• The curve inequality — Theorem 4.1. With separated weights, every algebraic curve in C^(m+1) has deg_W C >= (1 + sigma) times its total weighted contact with the logarithmic curves at the centres.
• Separated-weight interpolation — Theorem 1.1. With separated weights, polynomials of W-degree <= H realize every packet of jets of V-weight < H along the logarithmic curves at K centres with distinct coordinates, for all large H divisible by R.
• Ampleness on the blow-up — Proposition 4.3. On the blow-up of the jet ideal, A − E is ample: a positive multiple is a positive combination of the nef class from the curve inequality and an ample class.
• Direct images of twists on a blow-up — Lemma 5.1. For the blow-up p of a coherent ideal I, p_* O(−nE) = I^n and R^q p_* O(−nE) = 0 (q > 0) for all large n.
• Size of lcm(1, ..., n) — Lemma 2.1. log lcm(1, ..., n) <= 4 n log 2 for every n >= 1.
• Arithmetic lower bound — Proposition 2.2. A nonzero full-row minor of the interpolation matrix at Gaussian rational centres has log|Delta_H|/(MH) >= −(1 − bbar) − E_ar.
• Collision estimate — Lemma 4.1. A determinant of Taylor-coefficient rows of entire functions, with n_a rows testing the same function, is at most exp(−(log 2/4) sum n_a^2 + MH(E_hol + o(1))).
• Determinant comparison — Proposition 5.1. If g > 0, E_ar + E_an < g and c L > 1 + E_ar + E_an, approximations |pi − p_i/q_i| <= q_i^(−nu) with the interpolation hypotheses lead to a contradiction.
• The irrationality exponent of pi is 2 — Theorem 2.1. For every nu > 2 there is Q(nu) with |pi − p/q| > q^(−nu) for all integers p and q >= Q(nu); hence pi is irrational and mu(pi) = 2 (OpenAI, 2026).
• Spacing of multiples — Lemma 3.1. If ||q alpha|| >= c q^(1−kappa) for all q, the points q alpha, K <= q < 2K, are d-separated on the circle with d = c (2K)^(1−kappa).
• Generalized Flint–Hills series — Theorem 3.2. sum n^(−a) |sin n|^(−b) converges if and only if a > max(1, b); in particular sum 1/(n^3 sin^2 n) converges.

Reading order

Read the first lesson for the statement and the plan. The second and third lessons prove the zero estimate: a multiplicity estimate and the curve inequality for logarithmic curves. The fourth turns the curve inequality into interpolation through positivity on a blow-up; it uses the course on intersection numbers and positivity. The fifth lesson builds the interpolation determinant and proves its two bounds, and the last chooses the parameters, proves that the exponent is 2, and treats the Flint–Hills series.