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This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze more carefully about both the visible points and the invisible points in the definition of previous research. We prove asymptotic formulas for the number of lattice points in different levels of visibility.
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We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger than a positive constant.
Let $a,n in mathbb{Z}^+$, with $a<n$ and $gcd(a,n)=1$. Let $P_{a,n}$ denote the lattice parallelogram spanned by $(1,0)$ and $(a,n)$, that is, $$P_{a,n} = left{ t_1(1,0)+ t_2(a,n) , : , 0leq t_1,t_2 leq 1 right}, $$ and let $$V(a,n) = # textrm{ of visible lattice points in the interior of } P_{a,n}.$$ In this paper we prove some elementary (and straightforward) results for $V(a,n)$. The most interesting aspects of the paper are in Section 5 where we discuss some numerics and display some graphs of $V(a,n)/n$. (These graphs resemble an integral sign that has been rotated counter-clockwise by $90^circ$.) The numerics and graphs suggest the conjecture that for $a ot= 1, n-1$, $V(a,n)/n$ satisfies the inequality $$ 0.5 < V(a,n)/n< 0.75.$$
Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $mathscr B$-free numbers with Sarnaks conjecture on the `randomness of the Mobius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small height, curves with many integral multiples of a point, curves with high multiples of a point integral, and over two hundred curves with more than one hundred integral points. In addition, a novel and complete classification of self-descriptive numbers is constructed by bounding the number of zeros such a number must contain.
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