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In his 2006 ICM invited address, Konyagin mentioned the following conjecture:if $S_n f$ stands for the $n$-th partial Fourier sum of $f$ and ${n_j}_j\subset\N$ is a lacunary sequence, then $S_{n_j} f$ is a.e. pointwise convergent forany $f\in L\log\log L$. In this paper we will show that $| \sup_{j}|S_{n_j}(f)| |_{1,\infty}\leq C |f|_{1} \log\log(10+\frac{|f|_{\infty}}{|f|_1})\:.$ As a direct consequence we obtain that$S_{n_j}f \rightarrow f $ a.e. for $f\in L\log\log L\log\log\log L$. The(discrete) Walsh model version of this last fact was proved by Do and Lacey buttheir methods do not (re)cover the (continuous) Fourier setting. The keyingredient for our proof is a tile decomposition of the operator $\sup_{j}|S_{n_j}(f)|$ which depends on both the function $f$ and on the lacunarystructure of the frequencies. This tile decomposition, called$(f,\l)-$lacunary, is directly adapted to the context of our problem, and,combined with a canonical mass decomposition of the tiles, provides the naturalenvironment to which the methods developed by the author in "On the Boundednessof the Carleson Operator near $L^1$" apply.

On the pointwise convergence of the sequence of partial Fourier Sums along lacunary subsequences