There is a long-standing problem raised by Beurling and Wintner on completeness of the dilation system {φ(kx) : k = 1, 2, · · · } generated by the odd periodic extension on R of any φ ∈ L2(0,1). In this talk, we will introduce Kozlov's astonishing claims in 1950s and exhibit some results on the Beurling-Wintner problem for characteristic functions, especially the Kozlov completeness problem.