To analyze the set $E_4$, given by

$E_4 = \left\{ \frac{(-1)^n a}{n} + \frac{2b}{m} \, \middle| \, (n, m) \in (\mathbb{N}^*)^2 \right\}$

we will need to examine the behavior of the terms to determine the maximum, minimum, upper bound, and lower bound. Here’s the breakdown:

1. Expression Analysis:

   • The term $\frac{(-1)^n a}{n}$ oscillates between positive and negative values due to the factor $(-1)^n$. As $n$ increases, $\frac{a}{n}$ approaches zero.
   • The term $\frac{2b}{m}$ is always positive (assuming $b > 0$) and decreases towards zero as $m$ increases.

2. Upper Bound and Maximum:

   • The maximum of $E_4$ depends on the sign and values of $a$ and $b$.
   • For $n = 1$ and $m = 1$, the expression becomes $(-1)^1 \cdot a + 2b = -a + 2b$.
   • This is likely to be the largest value, as increasing $n$ and $m$ would decrease both terms in magnitude.

3. Lower Bound and Minimum:

   • The minimum of $E_4$ occurs for large $n$ and $m$, where both $\frac{a}{n}$ and $\frac{2b}{m}$ approach zero, potentially yielding values close to zero if $a$ and $b$ are positive.
   • If $a$ and $b$ are negative, the minimum may involve larger negative values depending on the specific signs and magnitudes.

4. Upper and Lower Bounds:

   • The upper bound is $-a + 2b$, assuming this is the maximum possible value based on $n = 1$ and $m = 1$.
   • The lower bound approaches zero as $n$ and $m$ increase if $a$ and $b$ are positive.

For a precise calculation, knowing the specific values of $a$ and $b$ would help further refine these bounds.

Would you like further elaboration on any of these points, or is there a specific value for $a$ and $b$ you'd like me to consider?

Here are some follow-up questions you might find helpful:

1. How do the values of $a$ and $b$ affect the bounds of $E_4$?
2. What happens to $E_4$ if $a$ or $b$ is zero?
3. How does the oscillating term $(-1)^n$ influence the ensemble's structure?
4. How would the bounds change if $n$ and $m$ were restricted to a smaller range?
5. Could we use limits to formally derive the bounds of $E_4$?

Tip: When working with sets defined by oscillating sequences, analyze the behavior for both even and odd terms separately to understand the full range.