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Source: arXiv v2, pp. 3--4, Section 3.1 up to Lemma 3.1.

Setup

Let

$$\mathcal A=(a_i+d_i\mathbb Z)_{1\le i\le n}, \qquad 1<d_1\le\cdots\le d_n,$$

and let $Q=[d_1,\ldots,d_n]$. Write

$$Q=\prod_{i=1}^Jp_i^{\nu_i}, \qquad p_1<\cdots<p_J, \qquad Q_j=\prod_{i=1}^jp_i^{\nu_i},$$

with $Q_0=1$. At stage $j$, reveal the residue classes whose modulus has largest prime factor $p_j$:

$$B_j=\bigcup_{\substack{1\le i\le n\\P^+(d_i)=p_j}} \{x\in\mathbb Z/Q\mathbb Z:x\equiv a_i\pmod {d_i}\}.$$

Let $\pi_j:\mathbb Z/Q\mathbb Z\to\mathbb Z/Q_j\mathbb Z$ be reduction and

$$F_j(x)=\{x':\pi_j(x')=\pi_j(x)\}.$$

Fix parameters $0\le\delta_j\le1/2$. Start with the uniform probability $\mathbb P_0$ on $\mathbb Z/Q\mathbb Z$. Assuming $\mathbb P_{j-1}$ is constant on each $Q_{j-1}$-fiber, set

$$\alpha_j(x)=\frac{|F_{j-1}(x)\cap B_j|}{|F_{j-1}(x)|}.$$

This number is constant on $F_{j-1}(x)$. Define the new point masses as follows. If $\alpha_j(x)<\delta_j$, put

$$\mathbb P_j(x)=\mathbb P_{j-1}(x) \frac{\mathbf1_{x\notin B_j}}{1-\alpha_j(x)}. \tag{1}$$

If $\alpha_j(x)\ge\delta_j$, put

$$\mathbb P_j(x)=\mathbb P_{j-1}(x) \begin{cases} \dfrac{\alpha_j(x)-\delta_j} {\alpha_j(x)(1-\delta_j)},&x\in B_j,\\[6pt] \dfrac1{1-\delta_j},&x\notin B_j. \end{cases} \tag{2}$$

If $\alpha_j=\delta_j=0$, the relevant fiber has no point in $B_j$, so the formally indeterminate first line of (2) is never evaluated.

Fiber-mass calculation

On a fiber with $\alpha=\alpha_j(x)<\delta_j$, a proportion $1-\alpha$ of the points remains and receives multiplier $(1-\alpha)^{-1}$. Hence (1) preserves the fiber's mass.

On a fiber with $\alpha\ge\delta_j$, the average multiplier in (2) is

$$\alpha\frac{\alpha-\delta_j}{\alpha(1-\delta_j)} +(1-\alpha)\frac1{1-\delta_j}=1.$$

Thus in both cases

$$\mathbb P_j(F_{j-1}(x))=\mathbb P_{j-1}(F_{j-1}(x)). \tag{3}$$

Induction over all fibers proves that every $\mathbb P_j$ is a probability measure. Membership in $B_j$ depends only on the residue modulo $Q_j$, while $\alpha_j$ depends only on the residue modulo $Q_{j-1}$. Equations (1)--(2) therefore also show that $\mathbb P_j$ is constant on $Q_j$-fibers.

For a function $f$ define

$$\mathbb E_jf=\sum_{x\in\mathbb Z/Q\mathbb Z}f(x)\mathbb P_j(x),$$

and, for every $1\le j\le J$, define

$$M_j^{(1)}=\mathbb E_{j-1}\alpha_j, \qquad M_j^{(2)}=\mathbb E_{j-1}\alpha_j^2.$$

The source prints $j=1,\ldots,J-1$ in this final definition, although its criterion and proof use the moments through stage $J$. The range $1\le j\le J$ is the necessary and consistent correction supplied here.