Perturbed Polynomial Powers and Bourgain Entropy Obstructions for Khintchin Averages

Let $a\ge 2$, let $p\in \mathbb{Z}[n]$ be eventually increasing and eventually non-negative, and let ${\lambda }_n=a^{p(n)}+f(n)>0$, where $f(n)\in \mathbb{Z}$. We prove, using Bourgain’s bounded entropy criterion, that if $\log (1+f(n)a^{-p(n)})$ is eventually non-zero and decays geometrically in absolute value, then $({\lambda }_n)$ is neither $L^{\mathrm{\infty }}$-Khintchin nor $L^1$-Khintchin. In particular, for every $c\in \mathbb{Z}\setminus \{0\}$, every positive tail of $(a^{p(n)}+c{)}_{n\ge 1}$ is non-Khintchin. The same conclusion applies to the standard examples $a^n+c$, $a^n+b^n$, and, whenever eventually positive, $a^n-b^n$, with $a\mathrm{\ne }b$. Thus these perturbations of geometric powers lie on the unstable side of the Khintchin problem. This gives a negative answer, in the translated-power case, to the question of Fan–Fan–Queffélec–Queffélec on the stability of translated powers.