Mizan R. Khan

On the maximal difference between an element and its inverse in residue rings

We investigate the distribution of n - M(n) where M(n) = max{|a - b|: 1 ≤ a, b < n - 1 and ab ≡ 1 (mod n)}. Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on n - M(n). We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study n - M(n) than a more traditional way using exponential sums.

On the Uniform Distribution of Inverses modulo n

In this expository note we use uniform distribution to clarify a result on the difference of an element and its inverse in (Z/nZ)*. We then explain why our remarks apply to some other settings. In doing so we state and prove a couple of folklore theorems on uniform distribution.

GEOMETRIC PROPERTIES OF POINTS ON MODULAR HYPERBOLAS

Given an integer n > 2, let Hn be the set H n ={(a,b): ab ≡ 1 (modn), 1≤a,b≤n-1} and let M(n) be the maximal difference of b-a for (a,b) ∈ h n . We prove that for almost all n, n — M(n) = O (n 1/2+o(1) ). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n .

White’s Theorem

We give an exposition of White’s characterization of empty lattice tetrahedra. In particular, we describe the second author’s proof of White’s theorem that appeared in her doctoral thesis (Rogers in Doctoral dissertation 1993) [7].

An Application of Modular Hyperbolas to Quadratic Residues

Abstract There are many elementary proofs of the classical result that −1 is a quadratic residue of an odd prime p if and only if p ≡ 1 (mod 4). In this note we prove this result by using the symmetries of a modular hyperbola. Consequently, our proof has a more geometric flavor than many of the other proofs.