On Waring's problem: beyond Freiman's theorem

Let $k_i\in \mathbb{N}$ $(i\ge 1)$ satisfy $2\le k_1\le k_2\le \dots$. Freiman’s theorem shows that when $j\in \mathbb{N}$, there exists $s=s(j)\in \mathbb{N}$ such that all large integers $n$ are represented in the form $n=x_1^{k_j}+x_2^{k_{j+1}}+\dots +x_s^{k_{j+s-1}}$, with $x_i\in \mathbb{N}$, if and only if $\sum k_i^{-1}$ diverges. We make this theorem effective by showing that, for each fixed $j$, it suffices to impose the condition $$\sum _{i=j}^{\mathrm{\infty }}{k}_i^{-1}\ge 2\log k_j+\mathrm{4.71.}$$ More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when $k\in \mathbb{N}$ and $s\ge 100(k+1{)}^2$, all large integers $n$ are represented in the form $n=x_1^k+x_2^{k+1}+\dots +x_s^{k+s-1}$, with $x_i\in \mathbb{N}$.