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DISTRIBUTION OF GAPS BETWEEN THE INVERSES $\mathrm{mod} q$

Published online by Cambridge University Press:  27 January 2003

C. Cobeli ,
M. Vâjâitu  and
A. Zaharescu
C. Cobeli
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 70700, Romania (ccobeli@stoilow.imar.ro; mvajaitu@stoilow.imar.ro)
M. Vâjâitu
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 70700, Romania (ccobeli@stoilow.imar.ro; mvajaitu@stoilow.imar.ro)
A. Zaharescu
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 70700, Romania (ccobeli@stoilow.imar.ro; mvajaitu@stoilow.imar.ro) Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green St., Urbana, IL 61801, USA (zaharesc@math.uiuc.edu)
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Abstract

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Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential.

AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69

Keywords

Poissonian distributioninversesexponential sums
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 1 , February 2003 , pp. 185 - 203
Copyright
Copyright © Edinburgh Mathematical Society 2003