frontiers / frontier
Additive combinatorics: Sidon sets and N(h,k) bounds
- id
- vfr_496956067dc5ad79
- license
- CC-BY-4.0
- findings
- 22
- accepted core
- 1
- contested
- 1
- links
- 0
- sources
- 22
- evidence
- 22
- avg conf
- 0.83
e33/33 · finding.asserted · reviewer:will · 2026-06-03 · null→e123
Findings dataset
machine packet · packet.jsonsha256:9b7f66bf8e69b86d…22 rows, every one a fold over the signed log. Scrub the event ruler and the table reduces to that prefix.
22/22 rows
| id | claim | grade | conf | verifier | witness | last event |
|---|---|---|---|---|---|---|
| vf_0baa77910460c953 | For h>=3 the analog of N(h,k) admits a polynomial in k upper bound via h-squared-dissociated sets, replacing the prior exponential construction. | unreviewed | 0.55 | — | — | asserted · 4w |
| vf_11e2e2a19e75887d | 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. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_203f4969695ff256 | 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. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_26b3e76e95e01f3e | 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. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_37afa7032e4b3d06 | 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}. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_3bde881fed68d7dd | The h-squared-dissociated set construction needed for the polynomial bound has diameter polynomial in k, replacing geometric components in the original construction. | under review | 0.40 | — | — | reviewed · 4w |
| vf_44e0fc831ac3e089 | Roth's theorem: any subset of {1,...,N} of density alpha > C / log log N contains a nontrivial 3-term arithmetic progression. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_5a1d047a8c85953a | 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|. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_63f4bcd4f4904a08 | 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. | unreviewed | 0.95 | — | — | asserted · 4w |
| vf_69c0db505746c28e | 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. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_7273c1823848f6e3 | 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. | unreviewed | 0.85 | — | — | span repaired · 3w |
| vf_8aa6c35eb4287a8f | Plunnecke-Ruzsa inequality: if |A+A| <= K|A|, then |kA - lA| <= K^{k+l} * |A| for all nonnegative integers k, l. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_9c462305947cb237 | Schoen-Shkredov: density alpha > exp(-c * (log N)^{1/4}) suffices for 3-APs via combined Fourier and Bohr-set decomposition methods. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_a4c54b1f9ff55bba | Szemeredi's theorem: any subset of the natural numbers with positive upper density contains arbitrarily long arithmetic progressions. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_c9ab5970cdbcb6af | 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… | unreviewed | 0.99 | — | — | asserted · 8d |
| vf_d018b0dfe8574a33 | Functions with large Gowers U^{s+1}-norm correlate with nilsequences of step s; this inverse theorem is the analytic engine behind quantitative Szemeredi for arbitrary k. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_d7781ce5f2c7330a | Bourgain's Fourier-analytic increment argument lowers the density threshold for 3-AP existence in {1,...,N} to alpha > C * (log log N)^2 / sqrt(log N). | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_e2252a5f8d8a5049 | Sanders' bound: density alpha > C * (log log N)^4 / log N is sufficient to guarantee a 3-term AP in any subset of {1,...,N}. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_e6443b35810cc321 | Ruzsa covering lemma: if |A+B| <= K|A|, then B is contained in the sumset A - A translated by at most K elements; the engine of small-doubling structure theorems. | unreviewed | 0.85 | — | — | asserted · 4w |
| vf_eee5b19763f3c625 | Ellenberg-Gijswijt extend Croot-Lev-Pach to prove a 3-AP-free subset of F_3^n has size at most 2.756^n, settling the cap-set problem up to a constant in the exponent. | unreviewed | 0.85 | — | — | span repaired · 3w |
| vf_f16e736879b4b42c | Bloom-Sisask break the 'log barrier' for Roth: density alpha > 1 / (log N)^{1+c} for an absolute c > 0 suffices for a 3-AP, the first sub-1/log N bound. | unreviewed | 0.85 | — | — | span repaired · 3w |
| vf_f58c1d68daec29c3 | Lindstrom's upper bound for Sidon-set size in {1,...,N} is sqrt(N) + N^{1/4} + 1, tightening the Erdos-Turan bound by a factor of sqrt(2). | unreviewed | 0.85 | — | — | asserted · 4w |