Shirandami 2023 dense forests constructed grids

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Victor Shirandami, Dense Forests Constructed from Grids. arXiv:2303.14719 (2023); Math. Z. 305 (2023), no. 1, Paper No. 15, doi:10.1007/s00209-023-03331-5. The arXiv record (https://arxiv.org/abs/2303.14719, read 2026-10-02) names the Creative Commons Attribution 4.0 license. The folder-name PDF is arXiv v2 of 12 July 2023; the statements and pages below are v2's, and the journal version was not compared.

A dense forest in R^n is a set every long enough line segment comes epsilon-close to, with visibility function V(epsilon) recording the required length. Theorem 1.1 (pp. 3-4) takes k >= 2 matrices M_1, ..., M_k in GL_n(R) and proves two conditions equivalent. The first is that no nonzero vectors v_1, ..., v_k, each with rationally dependent components, satisfy M_1 v_1 = ... = M_k v_k; equivalently, every direction u makes at least one velocity M_i^{-1} u rationally independent. The print omits "nonzero", which its proof (p. 8) assumes. The second is that the union of the grids M_i' Z^n + g_i is a dense forest for all translations g_i and all (M_1', ..., M_k') with the same image as (M_1, ..., M_k) in (R^* \ GL_n(R) / GL_n(Q))^k. This answers a question of Adiceam, Solomon and Weiss (2022). Theorem 1.2 (p. 4) complements it: with d = n - 1, k > d^2, fixed M_1, ..., M_k in GL_{d+1}(R) and delta > 0, for Haar-almost all rotations (R_1, ..., R_k) in SO(d+1)^k and all translations g_i, the union of the grids R_i M_i Z^{d+1} + g_i is a dense forest with V(epsilon) << epsilon^{-d-sigma_d(k)-delta}, with implied constant depending on d and delta, where sigma_d(k) = d^2(d+1)/(k - d^2). As k grows, sigma_d(k) tends to 0, so the bound approaches the optimal order epsilon^{-(n-1)}; one of the paper's main novelties is measuring genericity against several measures coming from the Iwasawa decomposition.

Source: https://arxiv.org/abs/2303.14719.

Results to transcribe.

• Theorem 1.1: For k >= 2 and M_1, ..., M_k in GL_n(R), there are no nonzero v_1, ..., v_k with rationally dependent components and M_1 v_1 = ... = M_k v_k exactly when every union of the grids M_i' Z^n + g_i is a dense forest, over all translations g_i and all (M_1', ..., M_k') with the image of (M_1, ..., M_k) in (R^* \ GL_n(R) / GL_n(Q))^k; answers a question of Adiceam-Solomon-Weiss.
• Theorem 1.2: For d = n - 1, k > d^2, M_1, ..., M_k in GL_{d+1}(R) and delta > 0, for Haar-almost all rotations (R_1, ..., R_k) in SO(d+1)^k and all translations g_i, the union of the grids R_i M_i Z^{d+1} + g_i is a dense forest with V(epsilon) << epsilon^{-d-sigma_d(k)-delta}, the implied constant depending on d and delta, where sigma_d(k) = d^2(d+1)/(k - d^2).
• Optimality remark: Any finite union of grids admitting a visibility function V must satisfy V(epsilon) >> epsilon^{-(n-1)}, so Theorem 1.2 comes arbitrarily close to optimal as k grows, since sigma_d(k) tends to 0.