  The Lieb-Schultz-Mattis (LSM) theorem and its descendants impose strong
constraints on the low-energy behavior of interacting quantum systems. In this
paper, we formulate LSM-type constraints for lattice translation invariant
systems with generalized U(1) symmetries which have recently appeared in the
context of fracton phases: U(1) polynomial shift and subsystem symmetries.
Starting with a generic interacting system with conserved dipole moment, we
examine the conditions under which it supports a symmetric, gapped, and
non-degenerate ground state, which we find requires that both the filling
fraction and the bulk charge polarization take integer-values. Similar
constraints are derived for systems with higher moment conservation laws or
subsystem symmetries, in addition to lower bounds on the ground state
degeneracy when certain conditions are violated. Finally, we discuss the
mapping between LSM-type constraints for subsystem symmetries and the anomalous
symmetry action at boundaries of subsystem symmetric topological (SSPT) states.
