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

Abstract

We investigate the distribution of n − M ( n ) n - M(n) where \[ M ( n ) = max { | a − b |   :   1 ≤ a , b ≤ n − 1 \ and\  a b ≡ 1 ( mod n ) } . M(n)=\max \left \{ \, \left | a-b\right |\ :\ 1 \leq a,b\leq n-1 \textrm {\ and\ } ab \equiv 1\pmod n\right \}. \] 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 ) 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 ) n - M(n) than a more traditional way using exponential sums.