Is it easy to find a lattice vector whose length is in a specific interval?

Question

Say $L$ is a lattice of ${\bf R}^n$ with rank $m$. Let $\alpha, \beta \ (\alpha<\beta)$ be positive real numbers. Set $A_{L}=\{{\bf x}\in L: \alpha\leq \|{\bf x}\|\leq \beta\}.$ It may be difficult to find an element of $A_{L}$ if for instance we consider $\beta=\lambda_1(L).$ Can we say anything about $\#A_L$ and is it easy to find elements of $A_L$(if there exists) for general $\alpha,\beta?$

Answer 1

It is very easy to find a lattice vector in $A_L.$ It is enough to choose an integer $\lambda$ in the set $\bigl{[} \lfloor \frac{\alpha}{\|{\bf v }\|}\rfloor ,\lfloor\frac{\beta}{\|{\bf v} \|}\rfloor \bigr{]}$ for some lattice vector ${\bf v}$ with $\|{\bf v}\|<\alpha.$ Then we choose the lattice vector $\lambda {\bf v}.$ So if it is easy to find lattice vectors with length $<\alpha,$ then is is easy to find lattice vectors in $A_L.$