Abstract:
We consider the inequality
$$
\bigl|\sqrt{n}+\sqrt{m}-x\bigr|<\Delta
$$
where $m$ and $n$ are natural numbers, $x$ is sufficiently large real number and $0<\Delta<\frac12$.
In the talk, we prove the formula for the number of solutions of such inequality of the type
$$
J(x,\Delta)\,=\,\frac{2}{3}x^{3}\Delta\,+\,O\bigl(x^{4/3}(\ln{x})^{7/2}\bigr),
$$
which is asymptotic for $\Delta\gg x^{-5/3}(\ln{x})^{7/2+\varepsilon}$, $\varepsilon>0$.