Additive combinatorics: Sidon sets and N(h,k) bounds

The h-squared-dissociated set construction needed for the polynomial bound has diameter polynomial in k, replacing geometric components in the original construction.

OEIS A309370 a(10) >= 66: a verified Sidon set of 66 distinct binary vectors in {0,1}^10 (componentwise integer addition into {0,1,2}^10; all 2211 pairwise sums a+b (a<=b) distinct), improving the live OEIS public lower bound a(10) >= 63 by +3 and the prior local best 65 by +1. Found by an Opus-4.8 Canopus-loop proposer (no-context arm, local search); re-verified from scratch by verify_construction.verify_sidon (frozen gate). Witness sha256:6a358e229f98b723…

A Sidon set in {1,...,N} of size O(sqrt(N)) exists, attaining the elementary upper bound for h=2 sumsets up to a multiplicative constant.

Behrend's construction yields subsets of {1,...,N} of size N * exp(-c * sqrt(log N)) containing no nontrivial 3-term arithmetic progression, the densest such known until 2020.

Singer's perfect-difference-set construction in projective planes yields Sidon sets in {1,...,N} of size sqrt(N) + O(N^{1/4}), matching the elementary upper bound up to lower-order terms.

Gowers gave the first quantitative proof of Szemeredi's theorem with effective bounds, introducing higher-order Fourier analysis (Gowers norms U^k) as the central tool.

B_h sets, the h-fold generalization of Sidon sets, have maximum density satisfying |A| <= (h! * N)^{1/h} + O(N^{1/(2h)}) in {1,...,N}.

Roth's theorem: any subset of {1,...,N} of density alpha > C / log log N contains a nontrivial 3-term arithmetic progression.

Freiman's theorem: a finite set A of integers with |A+A| <= K|A| is contained in a generalized arithmetic progression of dimension at most d(K) and size at most f(K) * |A|.

A Bohr neighborhood B(Lambda, rho) is the set of x in Z/NZ where |gamma * x / N| < rho for all gamma in Lambda; its dimension d = |Lambda| controls density |B| / N >= rho^d, enabling Fourier-analytic density-increment iterations.

The polynomial method bounds 3-AP-free subsets of (Z/4Z)^n by 4^{0.926n}, far below the 4^n total, breaking the previous logarithmic-style bounds for cap-set-style problems.

Plunnecke-Ruzsa inequality: if |A+A| <= K|A|, then |kA - lA| <= K^{k+l} * |A| for all nonnegative integers k, l.