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On the maximal difference between an element and its inverse in residue rings |
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| Authors: | Kevin Ford Mizan R. Khan Igor E. Shparlinski Christian L. Yankov |
| Affiliation: | Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 ; Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226 ; Department of Computing, Macquarie University, Sydney, NSW 2109, Australia ; Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226 |
| Abstract: | We investigate the distribution of  where  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  . 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  than a more traditional way using exponential sums. |
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