When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$

Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, and V is a subspace of $\mathbb{R^n}$.

Does this hold in general: ${\frac{d(L)} {d(L \cap V)}} \geq {\frac{d(M)}{d(M \cap V)}}$?

Why I am interested in this: There are analogies (analogue of the Riemann Roch theorem for example) which imply it makes sense to think of metrized lattices as sheaves and the log of d as the dimension of the global sections of a sheaf(by "metrized lattice" I mean a lattice with an inner product). So then given a map between metrized lattices, say f: A -> B, an analogous way to define "rank of induced map on global sections" would be: log(d(A)) - log(d(ker(f))) (from the rank-nullity theorem). I will denote this quantity as rank(f). So then if this rank function actually behaves as "rank of induced map on global sections", we would expect that $rank(g \circ f)$ is no greater than either rank(f) or rank(g). I am trying to show this for homomorphisms between metrized lattices that don't increase the norm at any point. $rank(g \circ f) \leq rank(f)$ is trivial; $rank(g \circ f) \leq rank(g)$ is equivalent to this problem I have posed.

I am interested in this rank function because it seems to be possible to do derived functor cohomology without an actual left-exact functor but with just the ranks of the maps that a left-exact functor would induce. Fully explaining myself here would result in an unreasonably big MO question.

$\textbf{Update:}$ The ideas mentioned in the previous paragraph, which are what originally motivated me to consider this question, are now fully explained here. Oded and Noah's result is applied in section 7.

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