Adiceam 2021 cut project quasicrystals lattices dense forests

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proposition_2_1: For lattices L_1, ..., L_s in R^N, some translates x_i + L_i have uniformly discrete union exactly when no difference set L_i - L_j is dense in R^N; the proof shows almost every choice of translates then works.

theorem_1_1: Every cut-and-project set in R^n misses the epsilon-neighborhood of some (n-1)-dimensional affine subspace for some epsilon > 0, so it is not a dense forest.

theorem_1_2: There exist uniformly discrete dense forests in R^2 that are finite unions of cut-and-project sets; the example is explicit, and the proof gives no bound on its visibility function.

theorem_1_3: Some union of three translated lattices in R^2 is a uniformly discrete dense forest with visibility v(eps) = O(eps^-(5+eta)) for every eta > 0; Section 6.1 writes the three lattices out explicitly.

theorem_1_4: For n >= 2, s >= n and eta > 0, almost every choice of ns lattices in R^n (in the paper's measure) has union a dense forest with v(eps) = O(eps^-(n-1+alpha_n(s)+eta)), where alpha_n(s) = n(n-1)^2/(s-(n-1)).

theorem_5_2: If an s-tuple of vectors in R^d is uniformly Diophantine of type Phi, the associated union of at most ns lattices in R^n, n = d+1, is a dense forest with v(eps) = O((eps^(d-1) Phi(d/eps)^-1)^d); for Peres's forest this gives O(eps^-3).

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Faustin Adiceam, Yaar Solomon, Barak Weiss, Cut-and-project quasicrystals, lattices and dense forests. Journal of the London Mathematical Society 105 (2022), 1167-1199. doi:10.1112/jlms.12534. arXiv:1907.03501. The copy read for this card is arXiv:1907.03501v2 (26 May 2021); the theorem, proposition, section and page numbers on this card refer to it.

Theorem 1.1 proves that a cut-and-project set in R^n is never a dense forest: it misses an epsilon-neighborhood of some affine hyperplane. Theorem 1.2 then gives an explicit uniformly discrete dense forest in R^2 that is a finite union of cut-and-project sets (the proof gives no visibility bound), and Theorem 1.3 gives three explicit lattices in R^2, suitably translated, whose union is a uniformly discrete dense forest admitting the visibility function O(eps^{-(5+eta)}) for every eta > 0; Theorem 1.4 shows that for each n >= 2, s >= n and eta > 0, almost every choice of ns lattices in R^n has union a dense forest with v(eps) = O(eps^{-(n-1+alpha_n(s)+eta)}), where alpha_n(s) tends to 0 as s grows. Separately, Theorem 5.2, applied to Peres's original three-lattice forest with the badly approximable golden ratio, improves Peres's O(eps^{-4}) visibility bound to O(eps^{-3}) (p. 18). The method combines torus-flow/homogeneous-dynamics formulations of the visibility condition with a uniform Diophantine condition on the lattice generators. Proposition 2.1 gives the exact pairwise criterion (no difference set L_i - L_j dense) under which translates of given lattices can be chosen with uniformly discrete union, and Section 6.1 supplies the explicit three-lattice example.

Source: https://arxiv.org/abs/1907.03501. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1907.03501), every other right reserved.

Bears on. #188: the paper does not mention the problem; its results concern sets that come within eps of every long segment, while the red class of a coloring as the problem asks must contain a point of every unit-step progression of the forbidden length and have no two points at distance 1. No result here gives such a coloring or a bound on that length.

Results. Labels and pages are those of arXiv:1907.03501v2.

• Theorem 1.1 (p. 2): a cut-and-project set in R^n misses the eps-neighborhood of some (n-1)-dimensional affine subspace, so it is not a dense forest.
• Theorem 1.2 (p. 3), with Theorems 2.4 and 2.5 (p. 9): uniformly discrete dense forests in R^2 that are finite unions of cut-and-project sets; explicit, with no visibility bound from the proof.
• Theorem 1.3 (p. 3), with Section 6.1 and Propositions 6.5-6.7 (p. 24): a union of three translated lattices in R^2 that is a uniformly discrete dense forest with v(eps) = O(eps^{-(5+eta)}) for every eta > 0; the lattices are explicit and the translations come from Proposition 6.5.
• Theorem 1.4 (p. 3), with Theorem 5.3 and Corollary 5.4 (p. 18): for n >= 2, s >= n and eta > 0, almost every choice of ns lattices in R^n, in the measure coming from a random s-tuple of vectors in R^{n-1}, has union a dense forest with v(eps) = O(eps^{-(n-1+alpha_n(s)+eta)}), alpha_n(s) = n(n-1)^2/(s-(n-1)).
• Theorem 5.2 (p. 18), with Definition 5.1 (p. 17): a uniformly Diophantine s-tuple of type Phi makes the associated union of at most ns lattices a dense forest with v(eps) = O((eps^{d-1} Phi(d/eps)^{-1})^d); with the golden ratio it gives Peres's forest the bound O(eps^{-3}), improving O(eps^{-4}).
• Proposition 2.1 (p. 7), with Corollary 2.2 (p. 7): translates of lattices L_1, ..., L_s in R^N can have uniformly discrete union exactly when no L_i - L_j is dense, and then almost every choice of translates works; a uniformly discrete union of two translated lattices is never a dense forest.

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