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I am interested in counting number of lattices using the following theorem. The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, corrected)" by J.W.S. Cassels:

Let $F(\bar{x})$ be a convex symmetric distance-function associated with a bounded convex set $$ \phi \ : F(\bar{x}) < 1 $$ of volume $$ V_F = V(\phi). $$ Let $\Lambda$ be a lattice with successive minima with $\lambda_1, ..., \lambda_n$ with respect to $F$. For real $t>0$ denote by $S(t)$ the set of $h \in \mathcal{R} / \Lambda$ which have at least one representative $\bar{y} \in t \phi \ (i.e. F(\bar{y})< t)$. Then the measure $m(S(t))$ of $S(t)$ satisfies the inequality \begin{eqnarray} m(S(t))) = \begin{cases} = t^n V_F \ \text{ if } t \leq \lambda_1/2 \\ \geq (\lambda_1/2) ... (\lambda_J/2)t^{n-J}V_F \ \text{ if } \lambda_{J}/2 \leq t \leq \lambda_{J+1}/2 \\ \geq (\lambda_1/2) ... (\lambda_n/2)V_F \ \text{ if } \lambda_{n}/2 \leq t. \\ \end{cases} \end{eqnarray}

Here $\mathcal{R}$ is the euclidean space which the lattice sits in.

I am interested in deducing information about $N(U) = \# \{ \bar{x} \in \Lambda : F(\bar{x}) < U \}$ when say $\lambda_{J} \leq U \leq \lambda_{J+1}$ using this theorem. I would like an upper bound and a lower bound for it. I am guessing it is possible, but I don't quite see how (I feel like there might be something very simple I am not aware of). I would greatly appreciate any assistance. I would also appreciate any references. Thank you very much!

PS I would also appreciate any reference or ways to estimate $N(U) = \# \{ \bar{x} \in \Lambda : F(\bar{x}) < U \}$ when $\lambda_{J} \leq U \leq \lambda_{J+1}$ not necessarily using the above theorem.

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