Question about linear transformations

[ythamsten]

Homework Statement

Hey PF, I'm here again asking about linear transformations, ha ha.
Let C={(x,y) $\in$ $\mathbb R$² | x²+y²≤1} a circle of radius 1 and consider the linear transform

T:$\mathbb R$²→$\mathbb R$²
(x,y) $\mapsto$ ($\frac{5x+3y}{4}$,$\frac{3x+5y}{4}$)​

Find all values of a natural n for which Tⁿ(C), the image of C after n applications of T, contains at least 2013 points (a,b) with coordinates a, b $\in$ $\mathbb Z$.

Homework Equations

N/A

The Attempt at a Solution

At first I've fixed for both input and output basis for the map ε = {e₁,e₂} (i.e. the canonic basis) writing the linear transformation in a matrix form:
$$\begin{pmatrix} T\end{pmatrix}^{ε}_{ε}\begin{pmatrix} x\\ y \end{pmatrix}_{ε} =\begin{pmatrix} \frac{5}{4} & \frac{3}{4}\\ \frac{3}{4} & \frac{5}{4} \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}_ε$$
Then, knowing I'm going to be applying the linear transformation n times, I thought would be a wise choice to diagonalize it, taking advantage of it's nice form to do so:
$$\begin{pmatrix} T\end{pmatrix}^{ε}_{ε} = -\frac{1}{2}\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}\begin{pmatrix} 2 & 0\\ 0 & \frac{1}{2} \end{pmatrix}\begin{pmatrix} -1 & -1\\ -1 & 1 \end{pmatrix}$$
But right now, I'm having a little bit of trouble to figure out how to count the number of points on the maps that has both coordinates belonging to $\mathbb Z$. Hope for some help, thanks in advance guys.

 

[Quesadilla]

I will respond, but I am not sure how much help I can offer. I have two questions, at least.

1. What does the image of a circle under a linear map look like?

2. We have spoken before about how the the determinant of the matrix associated with a linear map will tell you how area scales. So what is then the area of the transformed region after $n$ applications of the mapping?

After answering these questions, do you have any thoughts on it?

 

[ythamsten]

Well, I can see that after you apply the transformation on the circle, we expand vectors in the (1,1) by a factor of 2, and contract the ones in the (1,-1) direction by a factor of two either. Then, the image of the circle after applying T n times is limited by an ellipse centered at the origin, with semi-axis sized 2^n, in the (1,1) direction, and 1/(2^n), in the (1,-1) direction.
Since det(T)=1, applying the transformation doesn't actually change the value of the area.
My thoughts on it would be to count only the (c,c); c $\in$ $\mathbb Z$, since I'm sure that the number of those kind of points is growing with each time I apply T in the region C. Is this a good way to go?

 

[Quesadilla]

Right. It would seem to me as well that as $n$ grows larger the image looks more and more like the line $y=x$.

Maybe you can use your diagonalization to determine for which $n$ we reach points like $(1000,1000)$. You have $T^n \mathbf{x} = P^{-1}D^nP \mathbf{x}$, right?

Do you have access to an answer to the question?

 

[ythamsten]

Hm, I'm going to try that indeed... I guess the right answer would be n≥11. I think I would be able to find somewhere an answer model, since it was a question in a Math Olympiad. If you are interested let me now.