in our talk we will discuss the old and celebrated question regarding the pointwise behavior of fourier series near $l^1$. this presentation will include \begin{itemize} \item the resolution of konyagin's conjecture (icm, madrid 2006) on the pointwise convergence of fourier series along lacunary subsequences; \item the $l^1$-strong convergence of fourier series along lacunary subsequences. \item recent progress on the $l^1$-strong convergence of (the full) fourier series. \end{itemize} we end with several considerations on the relevance/impact of the above items on the subject of the pointwise convergence of fourier series.