Cardinality of the singular set in the binary additive problem with primes in short intervals (Q5960225)

From the introduction: In 1997, \textit{G. I. Arkhipov, K. Buriev} and \textit{V. N. Chubarikov} [Tr. Mat. Inst. Steklova 218, 28--57 (1997; Zbl 0915.11050)] obtained an upper estimate for the number \(T(x)\) of natural numbers \(n\) that do not exceed \(x\) and cannot be represented as the sum \(n= p+[\beta q]\), where \(p\) and \(q\) are prime numbers: a) \(T(x)\ll x^{1-\frac 29} \ln^8x\) if the incomplete partial fractions of the irrational number \(\beta\) are jointly bounded (condition A); b) \(T(x)\ll x^{1-\frac 29+\varepsilon}\) if \(\beta> 0\) is an irrational algebraic number (condition B), where \(\varepsilon>0\) is an arbitrarily small constant. In this paper these results are generalized to short intervals. Theorem. Let a number \(\beta>0\) satisfy either condition A or condition B and \(T(x,y)\) be the number of natural numbers \(n\) in the interval \((x-y,x)\) that cannot be represented in the form \(n=p+ [\beta q]\), where \(p\) and \(q\) are prime numbers. Then for \(x\to\infty\) and \((1-\varepsilon)x> y>(\ln x)^{\frac{9}{20}} x^{\frac{9}{10}} e^{-\frac{9}{10} c(\ln x)^{\frac 15}}\) the following estimate is valid: \[ T(x,y)\ll y^{-\frac{11}{9}}(\ln y)x^2 e^{-2c(\ln x)^{\frac 15}}, \] where \(c>0\) is some constant and \(\varepsilon>0\) is an arbitrarily small constant.