Approximations of the given real number by the sums of two square roots of integers.

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 $\Delta\gg x^{-5/3}(\ln{x})^{7/2+\varepsilon}$, $\varepsilon>0$.

^* Conference identificator: 947 3270 9056 Password: 555834