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The distribution of inverses in a residue ring modulo a given modulus. (English. Russian original) Zbl 0826.11032

Russ. Acad. Sci., Dokl., Math. 48, No. 3, 452-454 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 333, No. 2, 138-139 (1993).
For real \(\alpha\) let \(\{\alpha \}\) denote the fractional part of \(\alpha\), and \(|\alpha |= \min( \{\alpha \},1-\{ \alpha\})\). Let \(m\), \(n\), \(a\), \(b\) be integers and \(N\geq 1\). If \((m,n) =1\), let \(n_1\) be the inverse of \(n\pmod m\) with \(1\leq n_1< m\). Write \(X= (an_1+ bn)/m\) and \(Y= \exp(- \log^3 N/( 320 \log^2 m))\).
In this note the author gives the following results. Let \(m\geq m_0>1\), \((a,m) =1\), \(1\leq N\leq m^4\) and \(0\leq \alpha< \beta<1\). Let \(I\) denote the number of solutions of the system of inequalities \(\alpha\leq \{X\}< \beta\) and \(n\leq N\). Then \[ I\geq CN (\log N)^{- 3.5} ((\beta- \alpha)- Y) \] where \(C>0\) is an absolute constant.
As corollaries of the above results, the author further gives: (1) if \(Y< \beta- \alpha\), then the interval \([\alpha, \beta)\) contains a number of the form \(X\) with \(n\leq N\); (2) if \(\xi\) is a real number, then \(\min_{n\leq N} |\xi- X|\leq Y\).
The author claims that these results answer the questions posed in a conservation between him and C. Hooley (Maiori, Italy, September 1989).

MSC:

11J71 Distribution modulo one
11N69 Distribution of integers in special residue classes
11K06 General theory of distribution modulo \(1\)