All Questions
8 questions
17
votes
0
answers
1k
views
Almost monochromatic point sets
There are many sort of equivalent theorems about monochromatic configurations in finite colorings, such as Van der Waerden, Hales-Jewett or Gallai's theorem, the latter of which states that in a ...
- 18.8k
9
votes
0
answers
365
views
How to count integer lattice points close to a subspace of $\mathbb R^n$?
Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
- 686
7
votes
1
answer
642
views
volume over a hypercube, over simplex: twist by Euler numbers
Let $\square_n=\{(x_1,\dots,x_n): 0\leq x_i\leq1,\, \forall i\}$ be an $n$-dimensional unit hypercube, and let $\Delta_n=\{(u_1,\dots,u_n):u_1+\cdots+u_n\leq\frac{\pi}2,\, u_i\geq0,\, \forall i\}$ be $...
- 43.2k
11
votes
3
answers
2k
views
Could a perfect squared square be split into two perfect squared squares?
This is a geometric puzzle though it might conceivably
also define a special class of Pythagorean triples.
A perfect squared square PSS is a square (as a plane figure)
partitioned into smaller ...
- 1,375
1
vote
1
answer
210
views
Pythagorean triples related to non-isometric equidistant plane quadruples
QUESTION Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
$ x^2 + (x-u)^2\ =\ A^2$
$ x^2 + (x+u)^2\ =\ B^2$
?
...
- 7,500
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
14
votes
1
answer
837
views
Applications of the GCD metric
In the pre-MO era, I once realized that on the integers, the function
$$
d(m, n) := \sqrt{\log \frac{\sqrt{mn}} {\text{gcd}(m,n)}}\ ,
$$
is a metric (all properties are easily verified; in fact ...
- 28.6k
12
votes
3
answers
707
views
A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
- 139