  In this note, we propose gravity duals for 3+1 dimensional Lorentz invariant
theories exhibiting discrete scale invariance. We construct non-singular
solutions of a six dimensional gravitational theory that are warped products of
$AdS_{5}$ and a circle. The presence of non-trivial warp factor explicitly
breaks the symmetries of $AdS_{5}$ to discrete scale invariance and also breaks
translation symmetry along the circle. The matter content of the 6D
gravitational theory does not violate the null energy condition. In addition,
we show that the linearized fluctuations around these 6D backgrounds are stable
i.e., the fluctuations do not violate the Breitenlohner-Freedman stability
criterion. The dual theories display periodic RG trajectories, but these RG
flows do not violate the $a$-theorem. In particular, the dual field theory is
not a soft deformation (deformation by marginal or relevant operators) of a 3+1
dimensional conformal field theory and hence the $a$-theorem and similar
monotonicity constraints on RG flows do not apply to these examples. The
holographic $c$-theorem also does not apply to these solutions. Finally, we
present solutions of type II supergravity that exhibit discrete scale
invariance to demonstrate that such solutions can be embedded in string theory.
