OK. I think you can do this pretty easily by hand. First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A Sturmian sequence with parameter $\alpha$ has the property that sub-blocks of length $N$ have density converging to $\alpha$ uniformly in $N$.

Now: start by interspersing a Sturmian with parameter $\underline\delta$ with one of parameter $\bar\delta$. How to do the interspersing? Have one from $2^{n!}$ to $2^{(n+1)!}$. Then switch to the other between $2^{(n+1)!}$ and $2^{(n+2)!}$ etc. This switching is slow enough to guarantee that the sequence that you obtain has the prescribed $\underline\delta$ and $\bar\delta$. These are also the upper and lower densities for the time being.

Next we'll modify the sequence to obtain densities $\underline d$ and $\bar d$. Alternately splice in segments of Sturmian parameter $\underline d$ and $\bar d$ between $2^{4^i}$ and $2^{4^i+2^i}$. This won't affect the upper and lower logarithmic densities (because $2^i/4^i\to 0$).

But looking at these segments, you obtain a sequence with upper and lower densities $\underline d$ and $\bar d$ (the upper and lower uniform densities are also $\underline d$ and $\bar d$).

Finally, we'll perturb things as in my comment to get the uniform densities we want. For the segments between $2^n$ and $2^n+n$, insert alternately segments of the Sturmian sequences with densities $\underline u$ and $\bar u$. These segments are so sparse, they will have no effect on the upper and lower densities or the logarithmic densities. They are enough to guarantee that you get the uniform densities you want.

And Bob's your mother's brother.

OK. I think you can do this pretty easily by hand. First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A Sturmian sequence with parameter $\alpha$ has the property that sub-blocks of length $N$ have density converging to $\alpha$ uniformly in $N$.

Now: start by interspersing a Sturmian with parameter $\underline\delta$ with one of parameter $\bar\delta$. How to do the interspersing? Have one from $2^{n!}$ to $2^{(n+1)!}$. Then switch to the other between $2^{(n+1)!}$ and $2^{(n+2)!}$ etc. This switching is slow enough to guarantee that the sequence that you obtain has the prescribed $\underline\delta$ and $\bar\delta$. These are also the upper and lower densities for the time being.

Next we'll modify the sequence to obtain densities $\underline d$ and $\bar d$. Alternately splice in segments of Sturmian parameter $\underline d$ and $\bar d$ between $2^{4^i}$ and $2^{4^i+2^i}$. This won't affect the upper and lower logarithmic densities (because $2^i/4^i\to 0$).

But looking at these segments, you obtain a sequence with upper and lower densities $\underline d$ and $\bar d$ (the upper and lower uniform densities are also $\underline d$ and $\bar d$).

Finally, we'll perturb things as in my comment to get the uniform densities we want. For the segments between $2^n$ and $2^n+n$, insert alternately segments of the Sturmian sequences with densities $\underline u$ and $\bar u$. These segments are so sparse, they will have no effect on the upper and lower densities or the logarithmic densities. They are enough to guarantee that you get the uniform densities you want.

And Robert's your mother's brother.

OK. I think you can do this pretty easily by hand. First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A Sturmian sequence with parameter $\alpha$ has the property that sub-blocks of length $N$ have density converging to $\alpha$ uniformly in $N$.

Now: start by interspersing a Sturmian with parameter $\underline\delta$ with one of parameter $\bar\delta$. How to do the interspersing? Have one from $2^{n!}$ to $2^{(n+1)!}$. Then switch to the other between $2^{(n+1)!}$ and $2^{(n+2)!}$ etc. This switching is slow enough to guarantee that the sequence that you obtain has the prescribed $\underline\delta$ and $\bar\delta$.

Next we'll do the $\underline d$ and $\bar d$. Pick a fast increasing sequence $(n_i)$ and splice in segments of Sturmian parameter $\underline d$ and $\bar d$ between $2^{n_i}$ and $2^{n_i+\sqrt{n_i}}$. Provided the $n_i$ increase fast enough, this won't affect the upper and lower logarithmic densities (because $\sqrt{n_i}/n_i\to 0$).

But looking at these segments, you obtain a sequence with upper and lower densities $\underline d$ and $\bar d$ (the upper and lower uniform densities are also $\underline d$ and $\bar d$).

Finally, we'll perturb things as in my comment to get the uniform densities we want. For the segments between $2^n$ and $2^n+n$, insert segments of the Sturmian sequences with densities $\underline u$ and $\bar u$. These segments are so sparse, they will have no effect on the upper and lower densities or the logarithmic densities. They are enough to guarantee that you get the uniform densities you want.

And Bob's your mother's brother.

OK. I think you can do this pretty easily by hand. First you need a way to generate sequences with very uniform density. The Sturmian sequences are perfect for this. A Sturmian sequence with parameter $\alpha$ has the property that sub-blocks of length $N$ have density converging to $\alpha$ uniformly in $N$.

Now: start by interspersing a Sturmian with parameter $\underline\delta$ with one of parameter $\bar\delta$. How to do the interspersing? Have one from $2^{n!}$ to $2^{(n+1)!}$. Then switch to the other between $2^{(n+1)!}$ and $2^{(n+2)!}$ etc. This switching is slow enough to guarantee that the sequence that you obtain has the prescribed $\underline\delta$ and $\bar\delta$. These are also the upper and lower densities for the time being.

Next we'll modify the sequence to obtain densities $\underline d$ and $\bar d$. Alternately splice in segments of Sturmian parameter $\underline d$ and $\bar d$ between $2^{4^i}$ and $2^{4^i+2^i}$. This won't affect the upper and lower logarithmic densities (because $2^i/4^i\to 0$).

But looking at these segments, you obtain a sequence with upper and lower densities $\underline d$ and $\bar d$ (the upper and lower uniform densities are also $\underline d$ and $\bar d$).

Finally, we'll perturb things as in my comment to get the uniform densities we want. For the segments between $2^n$ and $2^n+n$, insert alternately segments of the Sturmian sequences with densities $\underline u$ and $\bar u$. These segments are so sparse, they will have no effect on the upper and lower densities or the logarithmic densities. They are enough to guarantee that you get the uniform densities you want.

And Bob's your mother's brother.