Pointwise convergence of Fourier series with respect to multiplicative systems

We study the pointwise convergence of Fourier series with respect to multiplicative Vilenkin systems. We derive some two-sided estimates of Dirichlet kernels. We find analogies of the Dini condition for the convergence of the Fourier series at some point $x$. In particular, we show that, whenever the condition
$$ \int_G\frac{|f(x\dotplus t)+f(x\overset{.}-t)-2f(x)|}{t}\,dt<\infty $$
guarantees the convergence of the Fourier series $f(x)$ at $x$ the same is not true of the condition
$$ \int_G\frac{|f(x\dotplus t)-f(x)|}{t}\,dt<\infty $$
(for unbounded systems).