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In this work we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space from the wave function maximum and relate the cumulants of this distribution to the properties of the quasi-local integrals of motion in the MBL phase. We demonstrate non-self-averaging property of the many-body fractal dimension $D_q$ and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of $D_q$ in the whole range of parameters. Introducing a simple toy-model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to non-ergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the non-interacting case.