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.$$ I am interest in enumerating (not just counting) all the points in $$Q_d \cap P_d.$$ Unfortunately I am not familiar with discrete mathematics/optimization and related subjects. …

Gauss Circle problem gives the number of lattice points lie within a circle of radius $r$. This question points to a reference that estimates the number of lattice points in a $d−$dimensional ball. … Is it possible to enumerate this number? If so, is there a known formula or procedure to compute or bound this number of lattice points? …

Moreover, I am not sure whether discrete subgroups of the isometry group are of particular interest because the application requires "good access" to the lattice points. … With generators of a discrete subgroup I am not sure I will be able to efficiently enumerate the lattice points in a manner that actually respects the distance between the lattice points. …

Then, $$L := \left\{ \sum_{i=1}^n v_i k_i, \text{ where } (k_1,\ldots,k_n) \in \mathbb{Z}^n \right\} $$ is a lattice. … I am interested in enumerating (not just counting) all the points in $L \cap B$ for $n$ up to dimension 12. Is this possible at all? Which algorithms are suited for this problem? …

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. … What other interesting problems could be solved or, at least in principle, be better understood by solving the above lattice point enumeration problem? References are certainly welcome! …

Consider lattice paths from $(0,0)$ to $(2n,2n)$ with steps $N=(0,1)$ and $E=(1,0)$ avoiding the points $(2i-1,2i-1)$ for all $1\leq i\leq n$. … A natural variation is: enumerate lattice paths (only north and east steps) from $(0,0)$ to $(2n-1,2n-1)$ avoiding the points $(2i,2i)$ for all $1\leq i\leq n-1$. …

Is it $\oplus P$ complete to decide if a convex body has odd number of integer points? … Update If you know the number of lattice points approximately then we can guess volume approximately. The converse is not true. What additional assumptions could give a healthy converse? …

I wonder if the various lattice-point theorems, such as Pick's Theorem or Minkowski's Lattice Theorem, have been generalized to the collection of points with rational coordinates no more than height … Here is a specific, Pick's Theorem -like question: Can the number $i$ of $h$-rational points inside a polygon $P$ be expressed in a form that is not tantamount to enumerating each interior point …

Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the rectangle … The problem posed above seems at least as hard as enumerating integer partitions, since for $x<m$ the number of such Young diagrams/lattice paths is precisely the number of integer partitions of $x$. …

One way of approaching this (for low dimensions $n$) might be to enumerate finite groups $G$ and their irreducible matrix representations. … Another thought is that it might be useful to restrict to the case where $G$ acts on a lattice, and then a natural choice might be to consider orbits consisting of short vectors of the lattice. …

To be precise, consider the square lattice $\mathbb{Z}^2$ as graph where the edges are pairs of points in the lattice having distance one from each other, where the distance is induced by the norm $\|( … We call a lattice animal the set of vertices of any connected subgraph of the square lattice. …

It turns out a depth first search to enumerate all possible lattice points $x\in C_q$ with $x$ elements bounded from above is very fast. … We use these recorded sets as a cache, and we further enumerate the lattice points in $C_q$. This also turns out to be fast since generally very few and often only one set overlaps with $C_q$. …

Points on an N-simplex" (Counting lattice points on an n-simplex) - the problem of finding solutions to expressions of the form: $a_1$$x_1$ + ... + $a_n$$x_n$ = S Is equivalent to counting the number … of integer lattice-points on (but NOT in the bounded area) the N-simplex being defined (i.e the real-number solution set to the expression). …

To every lattice point one may associate a Laurent monomial and therefore to every polytope we may associate the sum of all such monomials corresponding to lattice points inside the polytope. … Brion's formula allows one to write this Laurent polynomial as the sum of generating functions of the lattice points of the vertex cones of the polytope. …

The following, while probably impractical, should in theory do what you are asking: 1) Find a reduced lattice basis (LLL algorithm) 2) Use the reduced basis to enumerate all lattice points within a sufficiently … large box encompassing the point "P". 3) Compute the Voronoi diagram of this set of points. …