Convergence of series of translations

For a mean zero norm one sequence (f_n)L₂[0, 1], the sequence (f_n{nx+y}) is an orthonormal sequence inL₂([0, 1]²); so if, then converges for a.e. (x, y)[0, 1]² and has a maximal function inL₂([0, 1]²). But for a mean zerofL₂[0, 1], it is harder to give necessary and sufficient conditions for theL₂-norm convergence or a.e. convergence of. Ifc_n0 and, then this series will not converge inL₂-norm on a denseG subset of the mean zero functions inL₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c_n| = 0(n^–) for >1/2, then converges a.e. and unconditionally inL₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL_p[0, 1] with 1p<. Finally, if (f_n)L[0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x_n)[0, 1]. Moreover, iffL[0, 1] is mean zero and, then for a.e. (x_n)[0, 1], converges for a.e.y and in allL_p[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.