On the exceptional set in the generalized binary Goldbach problem (Q1594326)

Let \(\beta\) be irrational, and for \(x\to+\infty\) let \(T_1(x)\) denote the cardinality of the set of positive integers which are not representable as \(p+[\beta q]\), where \(p\) and \(q\) are primes, and \([\cdot]\) denotes the integral part. The authors prove that \(T_1(x)\ll_{\varepsilon} x^{1-2/9+\varepsilon}\) (though somewhat stronger bounds are true for suitable values of \(\beta\), depending on the structure of its continued fraction). Furthermore, the authors study a similar problem for the sum \(p+[q^c]\), where \(c>1\) is not integral, and obtain the corresponding bound \(T_2(x)\ll_{\varepsilon} x^{1-2\delta+\varepsilon}\) for the exceptional set, where \(\delta\) depends on the saving over the trivial bound in a suitable exponential sum. Actually this paper contains only the statement of the main results and of the most important lemmas (mainly estimates for exponential sums), without any reference to the complete proof.