32
votes
Accepted
Is there a regular pentagon with a rational point on each edge?
I'm considering a more generic problem, allowing points to lie on lines containing edges. Let use the $\mathbf p_i$ for the pentagon vertices and $\mathbf q_i$ for the rational points. The pentagon is ...
- 436
17
votes
Is there a regular pentagon with a rational point on each edge?
Update 3: The answer by uranix which finally settles the question is extremely clever! The essential point is the algebraic relation between $C, D, E$. The following Sage code (which can be run e.g. ...
- 22.5k
16
votes
Accepted
Counting primitive lattice points
No, there is no result in this form because in dimension 3 or higher it is allowed to have some non-first minima relatively small even when the first minimum is very small.
For example, for any $N>...
- 1,384
16
votes
Is there a regular pentagon with a rational point on each edge?
It is possible to modify Uranix's lovely answer to show that the only $n \ge 3$ for which there is a regular $n$-gon with a rational point on each side are $n=3,4,8$ (for which simple constructions ...
- 531
8
votes
Accepted
The number of quadratic forms attaining Hermite's constant
There are only finitely many inequivalent forms which may be local maxima for the Hermite invariant. Voronoi showed (1908) that the lattices attaining a local maxima are extreme, i.e. perfect and ...
- 3,209
8
votes
Accepted
Counting number of points on a lattice in a hypercube
After applying a suitable invertible linear transformation on $\mathbb{R}^n$, the lattice $\Lambda$ becomes $\mathbb{Z}^n$, and the box $[-X_1, X_1] \times \cdots \times [-X_n, X_n]$ becomes a ...
- 105k
7
votes
Sequential addition of points on a circle, optimizing asymptotic packing radius
This is really a comment, but it's getting a bit long for the comment box.
I want to point out that in addition to what you call the "greedy process," there's another obvious attempt, which could be ...
7
votes
Accepted
Counting matrices of bounded norm in SL_n(Z)
The order of growth is
$$
c_n X^{n^2 - n},
$$
where $c_n$ is a constant, explicit if we're working with the Euclidean norm. Thanks to Ofir Gorodetsky for pointing out the Duke-Rudick-Sarnak and ...
- 980
6
votes
Accepted
Sequential addition of points on a circle, optimizing asymptotic packing radius
I will show $\mu = \tfrac{1}{\log 4}$.
I first prove the upper bound $\mu \leq \tfrac{1}{\log 4}$. Fix a positive integer $r$. For $0 \leq k \leq r-1$, let $N_k$ be $2^{k/r} N$ rounded to the ...
- 156k
6
votes
Accepted
Number of planes generated by integer vectors
For $k=d-1$ this is a result of Bárány-Harcos-Pach-Tardos (2001). See Theorem 3 in the preprint version or the published version.
- 105k
5
votes
Siegel's Mean Value Theorem by Rogers and Macbeath
In this field it is a well known fact that Rogers had an error in his paper for the $n=2$ case. It was corrected by Schmidt in his paper A Metrical Theorem in the Geometry of numbers
- 51
5
votes
Sequential addition of points on a circle, optimizing asymptotic packing radius
Thinking more about Christian's stingy process, I have a new conjecture for the optimal $\mu$. I motivate the conjecture by modeling the evolution of the distribution of empty interval lengths in the ...
- 5,971
5
votes
Sequential addition of points on a circle, optimizing asymptotic packing radius
In this second answer, I want to discuss an upper bound on $\mu$ (not optimal, and I don't think this argument could give an optimal bound, even after fine tuning).
Suppose we have placed $N$ points ...
- 24.2k
5
votes
Accepted
Counting lattice points inside a parallelepiped
I have not filled in absolutely all the details, but hopefully this is enough to be convincing.
Let's let $M$ take positive integer values, and let's consider the parallelepiped: $$P'=\{x+Mtv\mid x\in ...
- 6,292
4
votes
Accepted
Lowering $i$th shortest vector of a lattice
There are lattices where your requirement cannot be met. In fact if you fix any lattice $\Lambda_0$ and you dilate it by a factor of $R$, then its determinant gets scaled by $R^n$, while its shortest ...
- 105k
4
votes
Accepted
Bounding the fractional parts of the $p^{\text{th}}$ roots of $n,n^2,...,n^{p-1}$
Let $n=m^p+1$ for some large enough $m$. For $0<k<p$ we then have $m^{kp}<n^k<m^{kp}+O(m^{(k-1)p})=m^{kp}(1+O(m^{-p}))$, so $$m^k<n^{k/p}<m^k(1+m^{-p})^{k/p}\leq m^k(1+O(m^{-p}))=m^k+...
- 28.2k
4
votes
Accepted
Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?
The answer is no.
By Proposition 2 of Gary Cornell and Michael I. Rosen 's paper The 𝓁-rank of the real class group of cyclotomic fields (paraphased):
Let $L/\mathbb Q$ be an abelian 𝓁-extension ...
- 9,501
4
votes
Accepted
Proof of generalized Siegel's mean value formula in geometry of numbers
Such a generalization (roughly) exists, known as Rodger's Integration Formula.
See Section 1.2 of Seungki Kim's Dissertation for a reference.
Theorems 1.2 and 1.3 are of interest.
Theorem 1.2: (...
- 2,183
3
votes
Finding a superbase in a lattice of Voronoi first kind
In the paper Finding a closest point in a lattice of Voronoi's first kind, the authors state in section 7 that:
Given a lattice, is it possible to efficiently decide whether it is of Voronoi’s first ...
- 2,183
3
votes
Maximal sublattice index in Minkowski's Second Theorem
For the case of $B$ a Euclidean sphere, the paper On the Index System of Well-Rounded Lattices is probably state-of-the-art (the well-rounded assumption is known to be WLOG). It investigates bounding ...
- 2,183
3
votes
Accepted
Bounds on Bézout coefficients
We may suppose that $-m<k\leqslant 0$ and choose integers $y_i$ such that $\sum y_ia_i=k$. Next, by replacing $(y_1,y_i)\to (y_1\pm a_i, y_i\mp a_1)$ we may achieve $y_i\in [0,a_1)$ for all $i>1$...
- 109k
3
votes
Accepted
Counting points on lattices in inside a box- Geometry of numbers
Let $\mathcal{S}\subset\mathbb{R}^n$ be a convex compact set lying in the closed ball of radius $R$ centered at the origin. Then
$$\left|\#(\mathcal{S}\cap\Lambda)-\frac{\mathrm{vol}(\mathcal{S})}{\...
- 105k
3
votes
Quadratic diophantine equations and geometry of numbers
Here is a basic (but not optimal) method for doing this.
Given a lattice $L$, a positive definite quadratic form $Q$ on $L$ and a bound $B$, you can enumerate all the elements $x\in L$ with $Q(x)\le ...
- 5,382
3
votes
Which lattices are rotatable into their scaled copy?
If one let F be a $d \times d$ matrix whose columns form an integral basis of $L$, $K \in O(d)$ be the rotation and $m>0$ be the scaling factor, then in order for $mKF$ to generate a sub-lattice, ...
- 487
2
votes
Intuition behind the proof of key step in Minkowski's second inequality on successive minima
As far as I can see it, there are two ideas behind $(\star)$:
First, one argues that it's enough to consider the volumes for $M_q^i$ instead of $M_q^n$. This is the case, because everything is ...
- 21
2
votes
Accepted
Minkowski's Linear Forms Theorem With Complex Coefficients
Credit where it's due to @so-calledfriendDon who found a source before I figured this out. However, I figure I'll post the proof because Google Books can be a finicky thing.
First, note that we can ...
- 201
2
votes
Bounds on Bézout coefficients
The following is likely useful, but doesn't answer your question really.
First, a standard (set of) inequalities used to bound the covering radius of a lattice is known by the name of transference.
...
- 2,183
2
votes
Accepted
Determinants of minors occurring 'within' determinant of full matrix
If the $n\times n$ matrix $M$ is decomposed into submatrices,
$$M=\begin{pmatrix}A&B\\ C&D\end{pmatrix},$$
where $A$ has dimension $m\times m$, then the determinant of $M$ can be decomposed as
...
- 188k
2
votes
Successive minima and the basis of lattice
Jacques Martinet has a paper that proves a result like this "on average".
For a lattice $\Lambda$, define $H_b(\Lambda) = \min_{\{v_i\}_i\text{ a basis of }\Lambda}\frac{\prod_{i}^n \lVert ...
- 2,183
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