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Visible lattice points along curves. (English) Zbl 1491.11088

A lattice point \((m,n) \in \mathbb{N}\times \mathbb{N}\) is said to be visible from the lattice point \((u,v) \in \mathbb{N}\times \mathbb{N}\), if there do not exist any other integer lattice points on the straight line segment joining \((m,n)\) and \((u,v)\). A historic result due to J. J. Sylvester [C. R. Acad. Sci., Paris 96, 409–413 (1883; JFM 15.0132.01)] says that the proportion of lattice points that are visible from the origin \((0,0)\) is \(1/\zeta(2)=6/\pi^{2}\).
In this present work the authors consider the distribution of lattice points which are visible from multiple base points simultaneously. They employ the definition that for any positive integer \(k\) and integer lattice points \((u,v)\) , \((m,n) \in \mathbb{N}\times \mathbb{N}\), where \(r\in \mathbb{Q}\) is given by \(n-v=r(m-u)^{k}\) and \(C\) be the curve \(y-v=r(x-u)^{k}\), then if there are no integer lattice points lying on the segment of \(C\) between points \((m,n)\) and \((u,v)\), then \((m,n)\) is defined to be Level-1 \(k\)-visible to \((u,v)\) . Furthermore, if there is at most one integer lattice point lying on the segment of \(C\) between points \((m,n)\) and \((u,v)\), then \((m,n)\) is said to be Level-2 \(k\)-visible to \((u,v)\) .
It follows that if a point \((m,n)\) is Level-1 or Level-2 \(k\)-visible to the point \((u,v)\) along the curve \(y-v=r(x-u)^{k}\), then \((u,v)\) is also Level-1 or Level-2 \(k\)-visible to \((m,n)\) , respectively, along the curve \(y-n= (-1)^{k+1}r(x-m)^{k}.\)
Let \(S\) be a given set of integer lattice points in the plane. The authors generalise the above definitions, and say that an integer lattice point \((m,n)\) is Level-1 \(k\)-visible to \(S\) if it belongs to the set
\(V_{k}^{1}(S) :=\) {\((m\) , \(n)\in \mathbb{N}\times \mathbb{N}\) : \((m , n)\) is \(k\)-visible to every point in \(S\) }.
Similarly, a point \((m,n) \in \mathbb{N}\times \mathbb{N}\) is defined to be Level-2 \(k\)-visible to \(S\) if it belongs to the set
\(V_{k}^{2}(S) :=\) {\((m\) , \(n)\in \mathbb{N}\times \mathbb{N}\) : \((m,n)\) is Level-2 \(k\)-visible to every point in \(S\) }.
For \(x\geq 2\), the authors consider visible lattice points along curves in the square \([\)1, \(x]\times[1,x],\) with the notation
\(N_{k}^{1}(S,x):=\#\{(m,n)\in V_{k}^{1}(S):m,n\leq x\},\)
and
\(N_{k}^{2}(S,x):=\#\{(m,n)\in V_{k}^{2}(S):m,n\leq x\}.\)
They focus on the interesting case when the points of \(S\) are pairwise \(k\)-visible to each other, so that the cardinality of \(S\) can’t be too large and is bounded by \(\# S\leq 2^{k+1}\). Their main results (Theorems 1.1 and 1.2) give asymptotic formulas for \(N_{k}^{1}(S,x)\) and \(N_{k}^{2}(S,x)\).

MSC:

11P21 Lattice points in specified regions
11M99 Zeta and \(L\)-functions: analytic theory

Citations:

JFM 15.0132.01
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Full Text: DOI arXiv

References:

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