  The Hilbert hotel is an old mathematical paradox about sets of infinite
numbers. This paradox deals with the accommodation of a new guest in a hotel
with an infinite number of occupied rooms. Over the past decade, there have
been many attempts to implement these ideas in photonic systems. In addition to
the fundamental interest that this paradox has attracted, this research is
motivated by the implications that the Hilbert hotel has for quantum
communication and sensing. In this work, we experimentally demonstrate the
fractional vortex Hilbert's hotel scheme proposed by G. Gbur [Optica 3, 222-225
(2016)]. More specifically, we performed an interference experiment using the
fractional orbital angular momentum of light to verify the Hilbert's infinite
hotel paradox. In our implementation, the reallocation of a guest in new rooms
is mapped to interference fringes that are controlled through the topological
charge of an optical beam.
