The concentration function of additive functions on shifted primes (Q1343727)

For additive functions \(f\), the author's aim is the study of two ``concentration functions'' \[ Q_ h= {1\over {\pi (x)}} \sum _{\substack{ p\leq x\\ h<f (p+a)\leq h+1 }} 1, \] where \(a\neq 0\) is a given integer, and \[ S_ h= {1\over {\pi (N)}} \sum _{\substack{ p\leq N\\ h<f(N- p)\leq h+1 }} 1. \] Defining \[ \begin{aligned} W(x) &= 4+ \min_ \lambda \Bigl( \lambda^ 2+ \sum_{p\leq x} {\textstyle {1\over p}} \min(1, | f(p)- \lambda \log p| )^ 2 \Bigr),\\ \text{and} Y(N) &= 4+\min_ \lambda \Bigl( \lambda^ 2+ \sum _{\substack{ p<N\\ (p, N)=1 }} {\textstyle {1\over p}} \min (1, | f(p)- \lambda \log p| )^ 2 \Bigr), \end{aligned} \] the author obtains the best possible upper estimates \[ Q_ h\;\ll\;W(x)^{-1/2} \tag{1} \] uniformly in \(h,f,x\geq 2\), and \[ S_ h \;\ll\;Y(N)^{-1/2} \tag{2} \] uniformly in \(h,f,N\geq 3\). Both of these equations are of the same quality as an estimate of \textit{I. Z. Ruzsa} [Acta Math. Acad. Sci. Hung. 36, 215-232 (1980; Zbl 0471.10034)] for the frequency of the integers \(n\), for which \(h< f(n)\leq h+1\). Equation (1) improves considerably on a result of \textit{N. M. Timofeev} [Acta Arith. 58, 113-131 (1991; Zbl 0726.11060)]. An application of (1) is the following: the number of \(p\leq x\), for which \(f(p+a)\) assumes a given value \(b\), is \(\ll \pi(x) \cdot E(x)^{- 1/2}\), where \[ E(x)= 4+ \sum _{\substack{ p\leq x\\ f(p) \neq 0}} {\textstyle {1\over p}}. \] An application of (2) shows that frequencies \(\nu_ N (p; f(N-p)\leq z)\) converge weakly to a distribution function if and only if the well-known three series condition of the Erdős-Wintner theorem is satisfied. The long and technically complicated proof of (1) connects \(Q_ h\) with a sum \(\sum_{n\leq x, (n, P_{y,z})=1} \dots\) of Fourier transforms of shifts of the multiplicative function \(g(m)= \exp (itf(m))\); ideas from Selberg's sieve method, the large sieve, and Bombieri-Vinogradov theorems for \(\alpha (n)= g(n)\log n- k_ 1 (n)- k_ 2 (n)\), \(\beta(n)= g(n)- \beta_ 1 (n)- \beta_ 2 (n)\), with suitably defined functions \(k_ j\), \(\beta_ j\), are used.