On a uniqueness theorem for the Franklin system

In this paper we prove that there exist a nontrivial Franklin series and a sequence$M_n$ such that the partial sums$S_{M_n}(x)$ of that series converge to 0 almost everywhere and $\lambda\cdot \mathrm{mes}\left\{x:sup_n\big|S_{M_n}(x)\big|>\lambda\right\}\to 0$ as $\lambda\to+\infty$. This shows that the boundedness assumption of the ratio $M_{n+1} /M_n$, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.