On the Uniqueness of Haar Series Converging over Subsequences of Partial Sums

We prove the existence of a sequence of positive integers $$n_i$$ , $$i\in\mathbb{N}$$ , that has zero density in $$\mathbb{N}$$ and possesses the following property: if the subsequence $$S_{n_i}(x)=\sum_{k=1}^{n_i}a_k\chi_k(x)$$ of partial sums of a Haar series converges everywhere to an everywhere finite integrable function $$f$$ , then this series is the Fourier–Haar series of $$f$$ .