Generalized Estermann's ternary problem for noninteger powers with almost equal summands

In the talk, we will discuss an asymptotic formula that generalizes Estermann's ternary problem for noninteger powers with almost equal summands. We will obtain an asymptotic formula for the number of representations of a sufficiently large positive integer $N$ as a sum of two primes and noninteger power of a positive integer.
Let $N$ be a sufficiently large positive integer, $\mathcal{L}=\ln{N}$, $c$ is a fixed noninteger number satisfying the following conditions:
$$ c\,>\,\frac{4}{3}+\mathcal{L}^{-0.3}, \qquad \|c\|\,\ge\,3c\bigl(2^{[c]+1}-1\bigr)\frac{\ln{\mathcal{L}}}{\mathcal{L}}. $$
Let $I(N,H)$ denote the number of solutions of the equation
$$ p_{1}+p_{2}+\bigl[n^{c}\bigr]\,=\,N,\quad \left| p_{i}-\frac{N}{3}\right|\,\le\,H, \quad i=1,2,\quad \left|\bigl[n^c\bigr]-\frac{N}{3}\right|\,\le\,H $$
in primes $p_1$, $p_2$ and a positive integer $n$. Then for $H\ge N^{1-\frac{1}{2c\mathstrut}}\mathcal{L}^2$ the following asymptotic formula is valid:
$$ I(N,H)\,=\,\frac{18}{3^{\frac{1}{c}}c} \cdot\frac{H^{2}}{N^{1-\frac{1}{c}}\mathcal{L}^2}+O\left(\frac{H^{2}}{N^{1-\frac{1}{c}}\mathcal{L}^3}\right). $$