I'm interested in particular representations of reals in residue number systems. Specifically, if we are given a real $0 \le n \le 1$, we wish to represent that number as a fraction in a residue number system. The fraction must be accurate to $m$ decimal places, but doesn't have to match $n$ exactly. We can pick any set of primes we like for the RNS, but their product must be $\ge r$. So, for example, if $r=30$, we could pick the primes as 2, 3, and 5, since $2 \cdot 3 \cdot 5 \ge 30$, or as 5 and 7, since $5 \cdot 7 \ge 30$. The hard part is that we can only represent $n$ using values that have multiplicative order $\le \alpha$, modulo the primes. So for instance, if we wish to approximate a number modulo a prime $p$, and we must use values with multiplicative order $\le \alpha$, we must be careful in which primes we choose. If $p=7, \alpha=2$, the only numbers modulo 7 that we can use are 1 and 6, since they are the only numbers with multiplicative order 2 or less.
I'm trying really, really hard to get a feel for the average case, when we are given $m$, $n$, $r$, and $\alpha$. Any help would be immensely appreciated. How can we find the smallest product of primes that this system allows, in the general case?
SOME THOUGHTS
I believe that we only have to concern ourselves with the denominators. If we are trying to represent a number modulo a prime $p$, we can first find the elements that have multiplicative order less than $\alpha$. Then find out what these elements (or values) represent as fractions. Continuing the example above, 6 modulo 7 is equal to 1/6, since $(6 * 6) \bmod 7 \equiv 1$. Further, 6 modulo 7 is equivalent to any fraction of the form $\frac{1}{6 + 7a}$, where $a$ is any integer. This is kind of like an equivalence class for fractions, with 1/7 of the fractions being represented by this single value, since we can make $a$ any integer we like. Further, 1 modulo 7 can represent any fraction of the form $\frac{1}{1 + 7a}$, so these 2 values represent approximately 2/7 of all fractions. I'm not even sure exactly where this gets us, but someone may be able to find a way to probabilistically examine the various residue equivalence classes and find what happens in the average case.