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Fix a discrete addition subgroup in $\mathbb{R}^n$. Given a finite spanning set, how can one find a group basis?

Fix a $d$-dimensional discrete addition subgroup $L\subset \mathbb{R}^n$. Call the spanning elements $v_1,\dots, v_k\in L$ and the matrix whose columns are these $v$'s as $V\in \mathbb{R}^{n\times k}$ …

Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$ With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the me …

Fix $U$ unitary over $\mathbb{R}^{K},$ take $U_n=I_{n\times n}\otimes U$ and denote the unit ball at 0 in $\mathbb{R}^n$ as $B^n$. For $d_1,\dots,d_K>0$, fix $S_n:=U_n\left(\prod_{k=1}^K d_k B^n\right …

The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?" For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\Omega(n …