9
$\begingroup$

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 subspace of $\mathbb R^n$ spanned by these vectors.

Informally, I want to count the "integer lattice points" (i.e., elements of $\mathbb Z^n$) whose Euclidean distance from $U$ is smaller or equal to some given number $a\geq0$.

More formally, if we consider the orthogonal projections onto $U$ of all elements of $\mathbb Z^n$ whose distance from $U$ is $\leq a$, then what I am looking for is an upper bound on the global density $\rho_U(a)$ in $U$ of these projection images ("density" is used here in the sense of number per volume in $U$).

Note that if we assume in addition that the vectors $v_1,...,v_m$ are themselves elements of $\mathbb Z^n$, then the distribution of projection images in $U$ will be repetitive (more precisely, $m$-fold periodic with periodicities $v_1,...,v_m$), so that in this case one only needs to count images contained in a "unit cell" with finite volume in $U$.

Now, since the density in $\mathbb R^n$ of lattice points $\mathbb Z^n$ is unity, I would heuristically expect (at least for large $a$) that the above density of projection images in $U$ behaves like the volume (in $\mathbb R^n$) of a "layer of thickness $a$ around $U$", so that \begin{align*} \rho_U(a) \approx V_{n-m}\ a^{n-m} \ , \tag*{(1)} \end{align*} where $V_{k} := \frac{\pi^{\frac{k}2}}{\Gamma(\frac{k}2+1)}$ denotes the volume of the $k$-dimensional unit ball. Using Stirling's approximation for the $\Gamma$ function, we can write (1) as \begin{align*} \rho_U(a) & \approx \frac1{\sqrt{\pi(n{-}m)}} \left( \!\frac{2\pi{\rm e}}{n{-}m} \!\right)^{\!\!\frac{n-m}2} \ a^{n-m} \\ & \leq \left( \!\frac{2\pi{\rm e}\, a^2}{n{-}m} \!\right)^{\!\!\frac{n-m}2} \ . \tag*{(2)} \end{align*} Clearly, inequality (2) cannot be valid in general for arbitrarily small $a$; for instance, if we choose $v_1,...,v_m \in \mathbb Z^n$ then the set of lattice points lying directly on $U$ (i.e. at zero distance) has finite density $\rho_U(0) > 0$ since it contains at least all linear combinations of $v_1,...,v_m$ with integer coefficients.

However, I believe that if we exclude all lattice points lying directly on $U$ from counting, an inequality of the form (2) may hold. More explicitly, let me try the following

Conjecture. For an arbitrary but fixed $m$-dimensional subspace $U$ of $\mathbb R^n$ ($1\leq m<n$) and for any $a \geq 0$, let $\rho_U(a)$ denote the density in $U$ of orthogonal projection images of all lattice points (i.e., elements of $\mathbb Z^n$) that lie at distances $\leq a$ from $U$. There is a universal constant $C$ such that for any choice of $m$, $n$, and $U$, and for all $a$: \begin{align*} \rho_U(a) - \rho_U(0) & \leq \left( \!\frac{C\, a^2}{n{-}m} \!\right)^{\!\!\frac{n-m}2} \ . \tag*{(3)} \end{align*}

My questions, of course, are:

  1. Is it known whether some inequality similar to (3) holds? Perhaps with restrictions on the choice of the subspace $U$ or the range of $a$?
  2. If the conjecture (or a similar one) is true, how can we prove it?
  3. If is is false, can we find a counterexample?

My feeling is that this problem ought to be solvable by relatively elementary arguments, but so far I have no clue how to approach it.

Addendum:

For the sake of clarity I will give a more formal definition of the quantity $\rho_U(a)$ here, assuming that $v_1,...,v_m \in \mathbb Z^n$, i.e. the vectors have only integer components.

Consider the orthogonal projection (in $\mathbb R^n$) of all elements of $\mathbb Z^n$ which lie at distances $\leq a$ from $U = {\rm span}(v_1,...,v_m)$ onto $U$. Let $S_a \subset U$ be the set of these projection points. The distribution of $S_a$ within $U$ is $m$-fold periodic with periodicities $v_1,...,v_m$, hence we can define the mean density of these points in $U$ as their mean density in a fundamental domain ("unit cell") $P \subset U$. An obvious fundamental domain is the $m$-parallelotope spanned by $v_1,...,v_m$, defined as \begin{align*} P := \big\{ \sum_{i=1}^m \alpha_i v_i : \forall_i\ 0 \leq \alpha_i \leq 1 \big\} \ . \tag*{(4)} \end{align*} The number of elements in the intersection of $S_a$ and $P$ is finite, so we can define the mean density in $P$ as this number divided by the volume of $P$: \begin{align*} \rho_U(a) := \frac{|S_a \cap P|}{{\rm vol}(P) } \ . \tag*{(5)} \end{align*} Note that if all components of $v_1,...,v_m$ are rational numbers instead of integers, one can proceed in exactly the same way, after multiplying them all with their common denominator (which doesn't change $U$). In contrast, if the components are arbitrary real numbers, generally there will be no periodicities and one has to resort to some limiting procedure to define $\rho_U(a)$.

However, currently I am mostly interested in the integer case $v_1,...,v_m \in \mathbb Z^n$ where definition (5) is sufficient.

$\endgroup$
  • 1
    $\begingroup$ The story starts at $(n,m)=(2,1)$, i.e., lines in the Euclidean plane. What do you know in this case? $\endgroup$ – YCor Feb 2 '18 at 16:02
  • $\begingroup$ In the case $(n,m)=(2,1)$ I believe that (3) is valid with any constant $C>4$. $\endgroup$ – Dierk Bormann Feb 2 '18 at 16:53
  • $\begingroup$ @AlexGavrilov $-$ $\rho_U(a)$ is supposed to be the average density of the projection points in $U$. I have tried to provide a suitable definition in an addendum. Hope this helps. $\endgroup$ – Dierk Bormann Feb 4 '18 at 19:28
  • $\begingroup$ @YCor $-$ Actually, I believe I can easily prove that for any $n\geq2$ and $m=n-1$, (3) is valid if $C>4$. Codimension $2$ or higher appears to be considerably harder. Still, it seems to me that for $(n,m)=(3,1)$, (3) is valid if $C>8$. $\endgroup$ – Dierk Bormann Feb 4 '18 at 19:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.