The author states in his introduction that “many classical arithmetical problems can be reduced to estimates for exponential sums, and the same philosophy in the reverse direction suggests that the estimation of exponential sums should be basically an arithmetical problem as well”.
Within the past nine years, two approaches to exponential sums have been introduced by E. Bombieri and H. Iwaniec [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 449-472 (1986; Zbl 0615.10047)] and the author [“Lectures on a method in the theory of exponential sums” (Tata Institute of Fundamental Research, Bombay, 1987; Zbl 0671.10031)]. A suitably chosen system for Farey fractions plays an important role in both methods.
In this article, the author proves the following theorem which should ultimately prove useful in exponential sum estimates.
Theorem. For \(2\leq K< K'\ll K\), let \({\mathcal K}\) be a set of integers in the interval \([K,K']\). Let \(B\gg 1\) and \(A\) be real numbers, and consider rationals \(h/k\) (in lowest terms) with \(k\in{\mathcal K}\) and \(h/k\in I= [A,A+B]\). Let \(L(x)\) be the number of these rationals that lie in the interval \(I(x)= [x- \Delta,x+\Delta]\), where \(K^{-2}\leq \Delta\leq B/2\).
Write \(\lambda=2 \Delta L/B\). Then \[ \int_ I (L(x)-\lambda)^ 2 dx\ll (B\Delta K^ 2+\Delta^ 3 K^ 4)(BK)^ \varepsilon. \tag{1} \] In addition, define \(L^*(x)= (2\Delta_ 0)^{-1} \int_{-\Delta_ 0}^{\Delta_ 0} L(x+u) du\) for \(\Delta\ll \Delta_ 0\leq \Delta/2\). Then (1) holds with \(L\) replaced by \(L^*\), and moreover \[ \int_ I \left( {{dL^*(x)} \over dx} \right)^ 2 dx\ll (B\Delta^{-1} K^ 2+ \Delta K^ 4) (BK)^ \varepsilon. \] The second term on the right of (1) comes from certain “end effects”; it can be omitted if \(\Delta \ll K^{-1}\). The first term can be interpreted by simple statistical heuristics.
For the entire collection see [Zbl 0772.00021].