Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

We determine the order of magnitude of H(x;y;z), the number of integers n x having a divisor in (y;z], for all x;y and z. We also study Hr(x;y;z), the number of integers n x having exactly r divisors in (y;z]. Whenr = 1 we establish the order of magnitude ofH1(x;y;z) for allx;y;z satisfying z x 1=2 " . For every r 2, C > 1 and " > 0, we determine the order of magnitude of Hr(x;y;z) uniformly for y large and y +y=(logy) log 4 1 " z min(y C ;x 1=2 " ). As a consequence of these bounds, we settle a 1960 conjecture of Erd} os and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.

http://annals.math.princeton.edu/wp-content/uploads/annals-v168-n2-p01.pdf

The distribution of integers with a divisor in a given interval

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

/pdf/the-distribution-of-free-path-lengths-in-the-periodic-4bqqzprvk8.pdf

The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \min(y^{C},x^{1/2-\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.

One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean

/pdf/artin-s-primitive-root-conjecture-a-survey-2otz5f739k.pdf

Artin's primitive root conjecture -a survey -

Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C

 1 boundary. For each integer Q≥ 1, we consider the integer lattice points from which are visible from the origin and prove that the 1
 st
 consecutive spacing distribution of the angles formed with the origin exists. This is a probability measure supported on an interval [m

 Ω,∞), with m

 Ω >0. Its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel.

Distribution of Lattice Points Visible from the Origin

A conjecture of R. R. Hall on Farey points

Let I = [α, β] be a subinterval of [0, 1]. For each positive integer Q , we denote by [Fscr ] I ( Q ) the set of Farey fractions of order Q from I , that is and order increasingly its elements γ j = a j / q j as α [les ] γ 1 2 < … < γ N I ( Q ) [les ] β. The number of elements of [Fscr ] I ( Q ) is We simply let [Fscr ]( Q ) = [Fscr ] [0,1] ( Q ), N ( Q ) = N [0,1] ( Q ). Farey sequences have been studied for a long time, mainly because of their role in problems related to diophantine approximation. There is also a connection with the Riemann zeta function which has motivated their study. Farey sequences seem to be distributed as uniformly as possible along [0, 1]; a way to prove it is to show that for all e > 0, as Q → ∞. Yet this is a very strong statement, as Franel and Landau [ 3 , 4 ] have shown that (1·3) is equivalent to the Riemann Hypothesis. Our object here is to investigate the distribution of spacings between Farey points in subintervals of [0, 1]. Various results related to this problem have been obtained by [ 2 , 3 , 5–8 , 10–13 ].

The h-spacing distribution between Farey points

Given a prime number p, Lehmer raised the problem of investigating the number of integers  for which a and  are of opposite parity, where  is such that . We replace the pair  by a point lying on a more general irreducible curve defined mod p and instead of the parity conditions on the coordinates more general congruence conditions are considered. An asymptotic result is then obtained for the number of such points.

Cristian Cobeli

Papers

Distribution of Lattice Points Visible from the Origin

A conjecture of R. R. Hall on Farey points

The h-spacing distribution between Farey points

Generalization of a problem of Lehmer

On the Distribution of Small Powers of a Primitive Root