divergent Fourier series in function spaces near $L^1[0;1]$

In this paper we generalize Bochkariev’s theorem, which states that for any uniformly bounded orthonormal system $\mathrm{\Phi }$, there exists a Lebesgue integrable function such that the Fourier series of it with respect to system $\mathrm{\Phi }$ diverge on the set of positive measure. We characterize the class of variable exponent Lebesgue spaces $L^{p(\cdot )}[0;1]$, $1<p(x)<\mathrm{\infty }$ a.e. on [0;1], such that above mentioned Bochkarev’s theorem is valid.