Kovrizhkin's lacunary Logvinenko–Sereda conjecture

Let $\Lambda=\{\lambda_k\}_{k=1}^{\infty}$ be a Zygmund lacunary set, and let $E$ be a measurable subset of $\mathbb{R}$. The Logvinenko–Sereda inequality is

$$\int_{\mathbb{R}} |f|^2 \leq C \int_E |f|^2$$

for functions satisfying $\widehat{f}\in L^2(F^c)$, where

$$F^c=\bigcup_{k=1}^{\infty}[\lambda_k,\lambda_k+1].$$

Kovrizhkin's conjecture. The Logvinenko–Sereda theorem holds for this lacunary set $F^c$.

Kovrizhkin proved the result under additional assumptions that the thickness parameter is sufficiently close to $1$ or that the intervals are sufficiently small; the conjecture asserts that neither additional condition is necessary.