{
  "id": "the-irrationality-exponent-of-pi",
  "title": "The irrationality exponent of π",
  "description": "The theorem that the irrationality exponent of π is 2: for every ν > 2, |π − p/q| > q^(−ν) for all large q. Irrationality exponents, Dirichlet's and Liouville's theorems; a weighted multiplicity estimate along commuting vector fields, proved with formal flow coordinates and Bézout's inequality; a curve inequality for logarithmic curves with separated weights, by a zero estimate and logarithmic residues; interpolation of jets on logarithmic curves through a blow-up, Kleiman's theorem and Serre vanishing; an interpolation determinant at Gaussian rational centres with an arithmetic lower bound and an analytic upper bound from Taylor collisions; and the convergence of the Flint–Hills series, with the exact convergence region of the sums of n^(−a)|sin n|^(−b).",
  "units": [
    {
      "id": "irrationality-exponents",
      "title": "Irrationality exponents",
      "reader": "irrationality-exponents.html",
      "source": "src/irrationality-exponents.md",
      "source_sha256": "edf9c294c876b698f3fcdbdb8ac071d1dd002161602f95d0d1ddcb2a7d8a48db",
      "status": "author_self_checked"
    },
    {
      "id": "weighted-multiplicity-estimates",
      "title": "Weighted multiplicity estimates",
      "reader": "weighted-multiplicity-estimates.html",
      "source": "src/weighted-multiplicity-estimates.md",
      "source_sha256": "9ccdb467e63412ea7c16230676e4ba78fac66ec8ab79330a75a23789976aaac5",
      "status": "author_self_checked"
    },
    {
      "id": "the-curve-inequality",
      "title": "The curve inequality",
      "reader": "the-curve-inequality.html",
      "source": "src/the-curve-inequality.md",
      "source_sha256": "ea7b41736565e67e00c076cc4125c12507009e79c26e0dde2c378493fa6db4f1",
      "status": "author_self_checked"
    },
    {
      "id": "interpolation-on-logarithmic-curves",
      "title": "Interpolation on logarithmic curves",
      "reader": "interpolation-on-logarithmic-curves.html",
      "source": "src/interpolation-on-logarithmic-curves.md",
      "source_sha256": "89d32b27d4e412180ba94e7cd284d38beebf1d58cc0edbe6fb57ea45aefd6d61",
      "status": "author_self_checked"
    },
    {
      "id": "an-interpolation-determinant-for-pi",
      "title": "An interpolation determinant for π",
      "reader": "an-interpolation-determinant-for-pi.html",
      "source": "src/an-interpolation-determinant-for-pi.md",
      "source_sha256": "61ca77842be445c14f2dec459ec920ab42d0ca1b76c41957eec3abdacf7ec2bc",
      "status": "author_self_checked"
    },
    {
      "id": "the-irrationality-exponent-of-pi-is-two",
      "title": "The irrationality exponent of π is two",
      "reader": "the-irrationality-exponent-of-pi-is-two.html",
      "source": "src/the-irrationality-exponent-of-pi-is-two.md",
      "source_sha256": "f428987a7eb1f76cbe2583a0214f75a021188897fbe1b36463ff4abad1ebcbd4",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "Course intersection-numbers-and-positivity (degrees on curves, Kleiman's theorem, weighted Bézout inequality)",
    "AG-MO lessons on blowing up, very ample sheaves and normalization",
    "AG-QC lessons on Serre's theorems and the Leray comparison",
    "AG-CA lessons on dimension theory, regular local rings and the Cohen structure theorem",
    "Core course C50 (Cauchy's estimates)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}