Linear transformation: Find the necessary quantity of T

[LCSphysicist]

Homework Statement

.

Relevant Equations

..

> Let $C$ be the disk with radius 1 with center at the origin in $R^2$.
> Consider the following linear transformation: $T: (x,y) \to (\frac{5x+3y}{4},\frac{3x+5y}{4})$
>
> What is the lowest number such that $T^{n}(C)$ contains at lest $2019$ points $(a,b)$, with a and b integers.


So we have $x²+y² = 1$,

$T: (x,y)\to ( x + y + \frac{x-y}{4}, x + y - \frac{x-y}{4})$

$T²: (x',y') \to ( x + y + \frac{3(x-y)}{8}, x + y - \frac{3(x-y)}{8})$

To be pretty honest, i couldn't see any pattern that simplify the solution of this problem...

 

[fresh_42]

You should find the matrix of $T$ and look how the volume of the disc changes, or the square of length two around the origin to start with an easier area.

 

[LCSphysicist]

You should find the matrix of $T$ and look how the volume of the disc changes.

$$
\begin{pmatrix}
x'\\ y'

\end{pmatrix}
=
\begin{pmatrix}
5/4 &3/4 \\
3/4 &5/4
\end{pmatrix}^{n}
\begin{pmatrix}
x\\ y

\end{pmatrix}$$

Apparently this changing of basis transform the circle (disc) in an ellipse. I think what you mean is to calculated the area of the new disc (ellipse)? (and not the volume?)

 

[fresh_42]

$$
\begin{pmatrix}
x'\\ y'

\end{pmatrix}
=
\begin{pmatrix}
5/4 &3/4 \\
3/4 &5/4
\end{pmatrix}^{n}
\begin{pmatrix}
x\\ y

\end{pmatrix}$$

Apparently this changing of basis transform the circle (disc) in an ellipse. I think what you mean is to calculated the area of the new disc (ellipse)? (and not the volume?)

Yes, volume is the general term for any dimension. Choose the unit square and see how the area will change with every step. What do you observe?