I am looking for a reference for the following statement:

For every integer $d$, there exist an integer $k$, such that for all polynomials $P_1, \ldots, P_k$ of degree $d$ there exist integers $N, q, a$, such that all integers $n>N$ satisfying $n\equiv a\pmod{q}$ can be represented as $n=P_1(x_1)+\dots+P_k(x_k)$, where the $x_i$ are integers in the range $\left[\left(\frac{N}{2s}\right)^{1/d}, \left(\frac{2N}{s}\right)^{1/d}\right]$, and $s$ is the sum of the leading coefficients of the $P_i$.

A lot of work has been done by many authors concerning additive problems with different monomials, Waring's problem with general polynomials, and strong localization of variables, but I haven't been able to find a reference for the above statement.

I am looking for a reference for the following statement:

For every integer $d$, there exist an integer $k$, such that for all polynomials $P_1, \ldots, P_k$ of degree $d$ there exist integers $N, q, a$, such that all integers $n>N$ satisfying $n\equiv a\pmod{q}$ can be represented as $n=P_1(x_1)+\dots+P_k(x_k)$, where the $x_i$ are integers in the range $\left[\left(\frac{n}{2s}\right)^{1/d}, \left(\frac{2n}{s}\right)^{1/d}\right]$, and $s$ is the sum of the leading coefficients of the $P_i$.

A lot of work has been done by many authors concerning additive problems with different monomials, Waring's problem with general polynomials, and strong localization of variables, but I haven't been able to find a reference for the above statement.