Results on lattices, field norms, residue orbits, caps and sampling, 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.

• Erdős's parabola — Proposition 1.2. The p points (x/p, (x^2 mod p)/p) have no triangle of area below 1/(2p^2).
• Power improvement (OpenAI) — Theorem 1.3. There are eta, c_1 > 0 such that Delta(n) >= c_1 n^(-2+eta) for every large n.
• Area of projected points — Lemma 2.1. The projections of three columns with positive third coordinates span a triangle of area |det A|/(2 u_3 v_3 w_3).
• Prime intervals — Lemma 4.3. For every large n there is a prime in (n, 2n].
• Index and covolume — Lemma 1.1. Z^n/MZ^n has |det M| elements; a full-rank sublattice of Z^n has index equal to its covolume.
• Successive lengths — Lemma 2.1. det L <= lambda_1 ... lambda_s <= s^(s/2) det L.
• Integral planes — Lemma 3.2. Lambda cap y^perp has determinant I|y|/g, where g generates y . Lambda.
• Fixed nonzero determinant (OpenAI) — Proposition 5.1. Matrices with rows in a lattice of index I, of length at most X >= lambda_3, and determinant t != 0 number at most C (log 2X)^2 X^6/I^2.
• A sum of determinants — Lemma 2.2. The norm of the 3 x 3 determinant over a field with r^d elements is a sum of T determinants of monomials of degree d.
• Determinant obstruction (OpenAI) — Lemma 3.2. Three digit columns with distinct labels have a determinant modulo h with no integer representative in [-tau, tau].
• Diagonal form — Lemma 1.1. A residue matrix with a unit entry is P diag(1, B^b, B^e) Q; its row lattice has index B^(b+e) and contains B^e Z^3.
• Orbit bound — Lemma 2.1. The orbit of C under SL_3 modulo h has at least (21/64) h^8/(D^3 E^2) elements.
• The smaller divisor of digit matrices — Corollary 3.2. E[D | labels] <= 2 for every triple of labels, and E[D 1_F] <= 6 r^(-d) for the event F that two labels coincide.
• A cap in a box — Lemma 1.1. The box {0, ..., w-1}^3 contains at least w^3/q points of F_q^3 with no three on a line.
• The random set (OpenAI) — Lemma 2.1. Three distinct points of V satisfy no integer relation with a coefficient vector of length at most H.
• Inclusion weights — Proposition 3.1. W(A) <= C for affinely independent residues, C q^3/s for rank at least two, and C (q^3/s)^2 always.
• Lifting identity — Lemma 2.1. Given labels and digits, the probability of a set of matrices is h^9/(N^9 |O|) times their total weight.
• Probability of a bad triple (OpenAI) — Corollary 3.2. Three samples form a bad triple with probability at most C (log 2N)^2 h/(N^3 r^d).
• Weighted zero-determinant count (OpenAI) — Proposition 1.1. Singular matrices with columns in the box, rows in Lambda and distinct projections have total weight at most C log(2N) N^6/I^2.
• Deletion — Lemma 1.2. Some outcome keeps n_r points whose triangles all have area at least tau/(16N^3).
• Size of n_r — Lemma 2.1. A_k r^alpha <= n_r <= D_k r^alpha with alpha = 45435k + 16, which gives eta = 2/alpha.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.