Simultaneous visibility in the integer lattice. (English) Zbl 1527.11074
Two lattice points are said to be visible from one another if there is no lattice point on the open line segment joining them. The classical problem regarding the asymptotic density of the set of lattice points, visible from the origin has been well studied, including results by Dirichlet, Montgomery and Lehmer, the latter of which proved that that the asymptotic density of the set of points of \(\mathbb{N}^k\), visible from the origin, is \(\frac{1}{\zeta(k)}\).
A related problem studied here by the authors is for a given a set \(S\subset \mathbb{Z}^k\), to examine \(V (S)\) the set of points of \(\mathbb{Z}^k\), visible simultaneously from all points of \(S\). Here they define the set \(S\) to be admissible if every two points in \(S\) are mutually visible and consider points in the box \[ B = J_1 \times J_2 \times\ldots\times J_k\subset \mathbb{Z}^k, \] where \(J_i = [M_i, M_i +L_i)\) for some integers \(M_i\) and \(L_i\), for \(1 \leq i \leq k\), with the assumption that \(L_1 \geq \ldots \geq L_k\).
Their main result is an improved upper bound on the error term which says that the number of points of \(B\), visible from \(S\), satisfies, as \(L_k =\min\{L_1, \ldots, L_k\} \rightarrow \infty\), \[ V (S,B) \leq L_1 \cdots L_k\prod_{p\in\mathcal{P}}\left (1-\frac{s(p)}{p^k}\right )+E, \] where \[ E = \begin{cases} O\left (\max\{L_1 \log^{3r}{L_2}, (L_1L_2)^{2/3+\varepsilon}\}\right ), \text{ for all } \varepsilon > 0, \qquad & k=2,\\ O\left (L_1 \ldots L_{k-1}\right ),\qquad & k\geq 3. \end{cases} \] Here \(\mathcal{P}\) is the set of prime numbers, and for \( p\in\mathcal{P}\) \[ \pi_p : \mathbb{Z}^k\to (\mathbb{Z}/p\mathbb{Z})^k, \] denotes the natural projection, with \[s(p) = |\pi_p(S)|.\] In the special case when \(L_1 = L_2 =\dots = L_k = L\), the authors note that \[ V (S,B) \leq L^k\prod_{p\in\mathcal{P}}\left (1-\frac{s(p)}{p^k}\right )+E, \] where \[ E = \begin{cases} O\left (L^{4/3+\varepsilon}\}\right ), \text{ for all } \varepsilon > 0, \qquad & k=2\\ O\left (L^{k-1}\right ),\qquad & k\geq 3. \end{cases} \] They then go on to deduce the Schnirelmann density of the set of visible points from some sets \(S\).
A related problem studied here by the authors is for a given a set \(S\subset \mathbb{Z}^k\), to examine \(V (S)\) the set of points of \(\mathbb{Z}^k\), visible simultaneously from all points of \(S\). Here they define the set \(S\) to be admissible if every two points in \(S\) are mutually visible and consider points in the box \[ B = J_1 \times J_2 \times\ldots\times J_k\subset \mathbb{Z}^k, \] where \(J_i = [M_i, M_i +L_i)\) for some integers \(M_i\) and \(L_i\), for \(1 \leq i \leq k\), with the assumption that \(L_1 \geq \ldots \geq L_k\).
Their main result is an improved upper bound on the error term which says that the number of points of \(B\), visible from \(S\), satisfies, as \(L_k =\min\{L_1, \ldots, L_k\} \rightarrow \infty\), \[ V (S,B) \leq L_1 \cdots L_k\prod_{p\in\mathcal{P}}\left (1-\frac{s(p)}{p^k}\right )+E, \] where \[ E = \begin{cases} O\left (\max\{L_1 \log^{3r}{L_2}, (L_1L_2)^{2/3+\varepsilon}\}\right ), \text{ for all } \varepsilon > 0, \qquad & k=2,\\ O\left (L_1 \ldots L_{k-1}\right ),\qquad & k\geq 3. \end{cases} \] Here \(\mathcal{P}\) is the set of prime numbers, and for \( p\in\mathcal{P}\) \[ \pi_p : \mathbb{Z}^k\to (\mathbb{Z}/p\mathbb{Z})^k, \] denotes the natural projection, with \[s(p) = |\pi_p(S)|.\] In the special case when \(L_1 = L_2 =\dots = L_k = L\), the authors note that \[ V (S,B) \leq L^k\prod_{p\in\mathcal{P}}\left (1-\frac{s(p)}{p^k}\right )+E, \] where \[ E = \begin{cases} O\left (L^{4/3+\varepsilon}\}\right ), \text{ for all } \varepsilon > 0, \qquad & k=2\\ O\left (L^{k-1}\right ),\qquad & k\geq 3. \end{cases} \] They then go on to deduce the Schnirelmann density of the set of visible points from some sets \(S\).
Reviewer: Matthew C. Lettington (Cardiff)
MSC:
| 11P21 | Lattice points in specified regions |
| 11N36 | Applications of sieve methods |
| 37A44 | Relations between ergodic theory and number theory |
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