Generalization of the ternary Goldbach problem for almost equal summands (Q1851521)

The main result of the paper is as follows. Let \(N\) be a sufficiently large positive integer. Then for \(U = N^{5/8}\log^cN\), where \(c\) is a constant, the equation \[ N=p_1+p_2+[\sqrt{2}p_3], \] where \(\frac{N}3 - U<p_i< \frac{N}3 + U\), \(i=1,2\), \(\frac{N}3 - U<[\sqrt{2}p_3]< \frac{N}3 + U\) is solvable in terms of primes \(p_1\), \(p_2\) and \(p_3\). Moreover, the number of solutions of this equation is given by the asymptotic formula \[ T(N,U) = \frac{2\sqrt{2}}{2}\, \frac{U^2}{\log^3N} + O\bigg(\frac{U^2}{\log^4N}\bigg). \] The solution of this problem is based upon methods of Arkhipov, Buriev, and Chubarikov and also uses estimates of trigonometric sums with prime numbers in short intervals by Tao.