Currier 2025 more pointsets many rich lines

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theorem_1_3: States Currier's theorem that for a nice basis Lambda and 0 < alpha <= 1/2 the product P of A_{n^alpha}(Lambda) and A_{n^{1-alpha}}(Lambda) determines Omega_Lambda(n^2/r^3) r-rich lines for every r <= C' n^alpha, with C' > 0 depending on d and Lambda.

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Gabriel Currier, More pointsets with many rich lines. arXiv preprint (2025). arXiv:2510.09769. The copy read for this card is arXiv:2510.09769v1 (10 October 2025); the theorem, section and page numbers on this card refer to it.

The paper produces new point-line configurations matching the Szemeredi-Trotter incidence bound (Theorem 1.1) and its r-rich-line form (Theorem 1.2, at most O(n^2/r^3) r-rich lines for r <= sqrt n). Theorem 1.3 is the main result: for a nice basis Lambda (a Z-independent set closed under products up to Z-combinations) and 0 < alpha <= 1/2, the Cartesian product P = A_{n^alpha}(Lambda) x A_{n^{1-alpha}}(Lambda) determines Omega_Lambda(n^2/r^3) r-rich lines for every r <= C' n^alpha, with C' > 0 depending on Lambda and its size d, so a single pointset is optimal for every r in that range. The method replaces the elementary number-theoretic analysis used by Erdos, Sheffer-Silier, and Guth-Silier with purely incidence-geometric arguments, which makes the proofs simpler and scales to generalized arithmetic progressions with bases from number fields of arbitrary degree; Section 2 derives sharp lower bounds for Theorem 1.1 as well. Theorem 1.3 subsumes the earlier constructions: alpha = 1/2 with Lambda = {1} recovers Erdos's grid and alpha = 1/2 with Lambda = {1, sqrt k} recovers Guth-Silier. The paper does not mention Erdős problems; for problem 102 the relation is stated under Bears on.

Source: https://arxiv.org/abs/2510.09769. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2510.09769), every other right reserved.

Bears on. #102: the paper does not mention the problem. Taking Lambda real, alpha = 1/2 and r = 4, Theorem 1.3 gives plane sets of Theta_d(n) points with at least c|P|^2 lines of more than three points each, for some c > 0 depending on Lambda, in which every line has O_d(n^{1/2}) points (an observation of the result page). This bounds h_c above at those sizes only for c below that constant, at the order of Erdős's integer grid; it gives no lower bound on h_c(n) and does not decide whether h_c(n) tends to infinity.

Results. Labels and pages are those of arXiv:2510.09769v1.

• Theorem 1.3 (p. 2): for a nice basis Lambda and 0 < alpha <= 1/2, P = A_{n^alpha}(Lambda) x A_{n^{1-alpha}}(Lambda) determines Omega_Lambda(n^2/r^3) r-rich lines for every r <= C' n^alpha, with C' > 0 depending on d and Lambda; the page also records the Section 2 (p. 3) derivation of incidence-sharp configurations for Theorem 1.1.

Theorems 1.1 and 1.2 (p. 1, Szemerédi-Trotter in its incidence and rich-line forms, for n^{1/2} <= m <= n^2 and r <= n^{1/2}) and Theorem 2.2 (p. 3, Beck) are recalled from the literature, and Proposition 2.3 (p. 3) is a lemma of the proof; none has a page here.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.