Primes in special intervals and additive problems with such numbers (Q1886108)

For integers \(l, r\) with \(1 \leq l \leq r\) and a non-integer real \(a > 1\), let \(\mathcal P_{r, a}\) denote the set of primes \(p\) such that \[ (l - 1) + rm \leq p^a < l + rm \quad \text{for some } m \in \mathbb Z. \] Set \(\delta = 10^{-7}\) and \(\gamma = 6 \cdot 10^{-11}\), and suppose that \(x \geq x_0(a)\), \(r \leq x^{\delta}\), and \(q \leq x^{\delta}\). The main result of the paper under review is the asymptotic formula \[ \sum_{p \in \mathcal P_{r, a}, \, p \leq x} \chi(p)\exp( 2\pi if(p)) = \frac 1r \sum_{p \leq x} \chi(p)\exp( 2\pi if(p)) + O( x^{1 - \gamma/a^2}(\log x)^3), \tag \(*\) \] where \(f(X) \in \mathbb R[X]\) is a polynomial of degree \(\leq 4\) and \(\chi(n)\) is a Dirichlet character modulo \(q\). The \(O\)-implied constant in (\(*\)) and the value of \(x_0(a)\) depend only on the distance from \(a\) to the nearest integer. The author uses (\(*\)) to derive an asymptotic formula for the counting function of \(\mathcal P_{r, a}\) and for the numbers of solutions of several additive problems with unknowns restricted to primes from \(\mathcal P_{r, a}\). For example, he obtains an asymptotic formula for the number of representations of a large odd integer as the sum of three primes from \(\mathcal P_{r, a}\), with \(r, a\) as above.