The valuative capacity of the set of sums of $d$-th powers

If $E$ is a subset of the integers then the $n$-th characteristic ideal of $E$ is the fractional ideal of $\mathbb{Z}$ consisting of $0$ and the leading coefficients of the polynomials in $\mathbb{Q}[x]$ of degree no more than $n$ which are integer valued on $E$. For $p$ a prime the characteristic sequence of $Int(E,\mathbb{Z})$ is the sequence ${\alpha }_E(n)$ of negatives of the $p$-adic valuations of these ideals. The asymptotic limit $\lim_{n\to \infty}\frac{\alpha_{E,p}(n)}{n}$ of this sequence, called the valuative capacity of $E$, gives information about the geometry of $E$. We compute these valuative capacities for the sets $E$ of sums of $\mathrm{\ell }\ge 2$ integers to the power of $d$, by observing the $p$-adic closure of these sets.