Residue classes are the atomic unit of my research. I recently observed that when I had set up the unions of residue class collections in the specific form below, there were redundant residue classes, no matter what integer values were selected for a, b, c and d.
$$ \bigcup_{x \in \mathbb{N}} [ax + b]^*_{(cx + d)} $$
Being a curious individual, I wanted to take a closer look and see what was happening, more algebraically.
As a refresher, my preference is $\mathbb{N} = [1, \infty) \cap \mathbb{Z}$ and the starred residue class indicates that $ax + b$ should be a least residue mod $cx+d$ for all values of $x \in \mathbb{N}$, $ 0 \le ax+b < cx+d$ and each residue class $[a]^*_m$ is endowed with the infimum $a+m$.
At first I was convinced that $[ax + b]_{(cx+d)} \supseteq [ay+b]_{(cy+d)}$ because $(cx+d) | (cy+d)$. This is true, but only consequentially.
Let $cx+d < cy+d$, such that $y = x + \Delta x$. Assume if you will that $c$ and $d$ are relatively prime to each other.
Because $(cx+d) | (cy+d)$, $(cx+d) | (cx + c\Delta x+d)$ and, $$\frac{cx +d + c\Delta x}{cx+d} = 1 + \frac{c \Delta x}{cx+d} \in \mathbb{N} $$
$$c \nmid cx + d, c\Delta x \mid cx+d \implies \Delta x \mid cx+d$$
When $c \mid d$, without any loss of information, $\Delta x \mid x + \frac{d}{c}$ which implies $\Delta x \mid cx + d$.
For today, I will be investigating more closely the case where $c$ and $d$ are either not relatively prime or one is a divisor of the other, because there is at least the following example where $(cx+d) \mid (cy+d) $ does not imply $[ax+b]^*_{(cx+d)} \supseteq [ay+b]^*_{(cy+d)}$:
$$\bigcup_{x \in \mathbb{N}} [x+1]^*_{(2x+4)} = \bigcup \{[2]^*_6, [3]^*_8, [4]^*_{10}, [5]^*_{12}, [6]^*_{14}, [7]^*_{16}, [8]^*_{18}, ... \}$$
Where although $[2]^*_6 \supset [8]^*_{18}$, $[2]^*_6 \nsupseteq [5]^*_{12}$. However, we can still observe that if $$x \in \bigcup_{x' \in \mathbb{N}}[x']^*_{(cx'+ d)}$$ then $[ax+b]^*_{(cx+d)}$ is redundant in the union where x is allowed to range over the integers from 1 to infinity, because we have $c(x + n \Delta x) + d = c(x + n(cx +d)) + d $,
where $ x + n(cx +d) = \bigcup_{x \in \mathbb{N}}[x]^*_{(cx+ d)} \subset \mathbb{N}$
$$ay+b \equiv ax+b \pmod {cx+d} \implies a(x + n(cx +d))+b \equiv ax+b \pmod {cx+d} $$ $$ ax + b + an(cx +d) \equiv ax+b \pmod {cx+d}$$
So I want to know if this is a sufficient rough proof of the existence of redundant residue classes in "weak" sieves of the above form and how exactly to find them.