Intervals between numbers that are sums of two squares and Jacobi-type forms

Let $S$ be the set of all numbers that can be expressed as the sum of two squares and $R(x)$ be the distance from positive real number $x$ to the set $S4. Classical conjecture due to Euler states that $R(x)=x^o(1)$.
In my talk, I will show that Cohen--Kuznetzov construction of Jacobi-type forms can be used to establish new bounds for moments of $R(x)$.