How to generate uniform distributed samples with given auto-correlation function As I mentioned in the question title, I want to generate specific uniformly distributed samples.
I need them to model a real world scenario. For my real data, I estimated a function, which approximates the auto-correlation function by a e-function. I also modeled a distribution model for my real data.
I can generate samples, which follow my distribution model with the inverse transform sampling.
If i think correct, instead of "standard" uniform distributed samples, which i feed into the inverse transform sampling, i need to adapt those uniform distributed samples, so that their auto-correlation function follows my estimated e-function.
Now the question is, if i am correct, how to adapt my uniform distributed samples, that they correspond to my requirements.
It would be great if, somebody could help.
EDIT 1:
Here is a plot of the auto-correlation function of my real data (different scenarios) and the modeled function (green). The ACF of my uniform distributed samples should follow the green (respectively blue) line.

EDIT 2:
Thank you so much so far for the answers. To make my problem a bit clearer, i added a illustration of the problem. For a simulation, i need to generate samples, which follow my distribution model (with inverse transform sampling a easy task) AND have the same auto-correlation behavior like the real data.
My idea was, that i need to adapt the uniformly distributed samples, which i feed into the inverse transform sampling, so that their ACF follows my model.
The problem is, that i don't know if this idea is correct and that i don't know how to adapt the uniformly distributed samples.


Letting $U_t=\Phi(X_t)$ where $X_t$ is a zero-mean and unit-variance stationary Gaussian process with autocorrelation function $\rho_h$ and $\Phi$ is the standard normal cdf, it follows that each $U_t$ is marginally uniformly distibuted.  It follows that the relation between $\rho_h$ and the autocovariance function of $U_t$ is
\begin{align}
\gamma_h&=\operatorname{Cov}(U_t,U_{t+h})
\\&=E(U_tU_{t+h})-E(U_t)E(U_{t+h})
\\&=E(\Phi(X_t)\Phi(X_{t+h}))-1/4
\\&=P(Z_1\le X_t\cap Z_2\le X_{t+h})-1/4
\\&=P\left(\frac{Z_1-X_1}{\sqrt{2}}\le0\cap \frac{Z_2-X_{t+h}}{\sqrt{2}}\le 0\right)-1/4
\\&=\Phi_2\left(\begin{bmatrix}0\\0\end{bmatrix};\frac{\rho_h}2\right)-1/4.
\end{align}
Here $Z_1$ and $Z_2$ are independent standard normal random variables, and $\Phi_2$ is the cdf of the standard bivariate normal distribution with correlation $\rho_h/2$.
Using this result, the above relation simplifies to
$$
\gamma_h = \frac1{2\pi}\operatorname{arcsin}\frac{\rho_h}2.
$$
Solving this equation, we find that the autocorrelation of $X_t$ needed to achieve a target autocovariance $\gamma_h$ of $U_t$ is
$$
\rho_h = 2\sin(2\pi \gamma_h).
$$
Even if the target autocovariance function $\gamma_h$ is positive semi-definite, the above construction may not be feasible.  For example, if the target autocovariance function is that of a MA(1)-process with a unit root,
$$
\gamma_h=\begin{cases}
1/12 &\text{for }h=0 \\
1/24 &\text{for }h=1 \\
0 &\text{for }h>1
\end{cases},
$$
this would imply that
$$
\rho_h=\begin{cases}
1 &\text{for }h=0 \\
0.518 &\text{for }h=1 \\
0 &\text{for }h>1
\end{cases}
$$
which is not a positive semi-definite autocorrelation function.
