Analytic-function extension of the polynomial Koksma–Weyl theorem

Let $g:\mathbb{R}\to(1,\infty)$ be a nonconstant analytic function, let $b:\mathbb{N}\to\mathbb{R}$ tend to infinity and be scattered, let $k\in\mathbb{N}$, let $p_1,\dots,p_k:\mathbb{R}\to\mathbb{R}$ be nonconstant analytic functions, and let $a_1,\dots,a_k:\mathbb{N}\to\mathbb{R}$ be scattered. Assume that either the sequences $a_1,\dots,a_k$ are jointly scattered or the functions $1,p_1,\dots,p_k$ are linearly independent. Analytic Koksma–Weyl conjecture. The conclusion of the polynomial Koksma–Weyl theorem should still hold: for almost every $x\in\mathbb{R}$, the sequence $\big(g(x)^{b(n)},a_1(n)p_1(x),\dots,a_k(n)p_k(x)\big)$ is uniformly distributed in $\mathbb{T}^{k+1}$. This would extend the stated theorem from nonconstant polynomials to general nonconstant analytic functions; the paper's techniques do not establish it.