Bounded prime rank

---

The later one-unbounded-prime-power theorem allows arbitrary exponents at one distinguished prime per modulus and bounded complementary $\Omega$. In particular it proves natural density for arbitrary singleton residues with $\omega(q)\le2$, without coherence. The subsequent squarefull-kernel theorem extends this to all bounded-$\omega$ singleton systems, and more generally to bounded numbers of prime factors occurring to exponent exactly one.

Theorem: proved

Write $\Omega(q)$ for the number of prime factors of $q$, counted with multiplicity. If $\sup_i\Omega(n_i)<\infty$, the delayed survivor set in Problem 25 has natural density, equal to its finite-periodic candidate $\delta$. More precisely, the delayed excluded union has finite-periodic approximation in upper natural density.

The proof actually permits several residues at each modulus. It therefore does not address the distinctive difficulty of unbounded-rank singleton families. Duplicate congruence pairs can be discarded, so each modulus has only finitely many distinct normalized residues. No theorem from an external source is required below.

A finite counting and harmonic estimate

Let $C_i=[a_i]_{q_i}$ be finitely many full congruence classes, with $0\le a_i<q_i$, and let

$$B_i=C_i\cap[q_i,\infty),\qquad B=\bigcup_iB_i, \qquad \beta=\mu\left(\bigcup_iC_i\right),$$

where $\mu$ is Haar probability on $\widehat{\mathbb Z}$, equivalently periodic density for this finite family. Define

$$\kappa(q)=\sum_{d\mid q}\frac{\varphi(d)}d =\prod_{p^e\Vert q}\left(1+e(1-1/p)\right).$$

Then, for every $X\ge1$,

$$|B\cap[1,X]|\le \beta X\max_i\kappa(q_i), \qquad H_B(X)\le \beta H_{\lfloor X\rfloor}\max_i\kappa(q_i). \tag{1}$$

In particular $\Omega(q_i)\le r$ gives the constant $2^r$, since $\kappa(q)\le\tau(q)\le2^{\Omega(q)}$.

Proof. Partition the actually deleted integers among their first covering classes, obtaining disjoint sets $E_i\subseteq B_i$. In the counting version put $c_i=|E_i\cap[1,X]|/X$ and $\alpha=\sum_i c_i$. For every divisor-face $[b]_d$ put

$$S_{d,b}=\sum_{\substack{i:d\mid q_i\\a_i\equiv b\pmod d}}c_i.$$

All points counted by this sum lie in the delayed class $\{n\ge d:n\equiv b\pmod d\}$: activation gives $n\ge q_i\ge d$. Consequently $S_{d,b}\le1/d$.

On Haar space set $Z=\sum_i q_i c_i\mathbf1_{C_i}$. Then $\mathbb EZ=\alpha$ and $Z$ vanishes off a set of measure $\beta$. Compatible pairs of classes have intersection measure $1/\operatorname{lcm}(q_i,q_j)$; incompatible pairs have empty intersection. Using $\gcd(q_i,q_j)=\sum_{d\mid q_i,q_j}\varphi(d)$,

$$\begin{aligned} \mathbb EZ^2 &=\sum_{\substack{i,j:\ a_i\equiv a_j\pmod{\gcd(q_i,q_j)}}} c_ic_j\gcd(q_i,q_j)\\ &\le\sum_d\varphi(d)\sum_{b\bmod d}S_{d,b}^2\\ &\le\sum_d\frac{\varphi(d)}d\sum_{b\bmod d}S_{d,b} =\sum_i\kappa(q_i)c_i. \tag{2} \end{aligned}$$

The first inequality can be strict: an incompatible pair may agree modulo a proper divisor of its gcd and hence contribute to the right side. Cauchy--Schwarz now gives $\alpha^2\le\beta\mathbb EZ^2\le\beta(\max_i\kappa(q_i))\alpha$, proving the counting statement.

For the harmonic statement instead take $c_i=H_{E_i}(X)/H_{\lfloor X\rfloor}$. The same argument applies because

$$\sum_{\substack{n\le X\,,\ n\ge d\\n\equiv b\pmod d}}\frac1n \le\frac1d H_{\lfloor X/d\rfloor} \le\frac1d H_{\lfloor X\rfloor}.$$

Thus (2) and Cauchy--Schwarz remain valid. $\square$

The more informative form of (2) is $\alpha^2\le\beta\sum_i\kappa(q_i)c_i$; replacing its weighted factor by a universal constant is not justified.

Proof of the theorem

We prove by induction on $r$ that every family of congruence pairs with $\Omega(q)\le r$ has finite-periodic approximation in upper natural density. Multiple distinct residues per modulus are allowed in the inductive statement. The case $r=0$ has only modulus $1$ and is immediate.

Let $U$ be the union of the full cylinders and $\beta=\mu(U)$. If $\beta=1$, finite truncations already force the delayed excluded union $B$ to have natural density $1$. If $F$ is a finite subfamily, then $d(B\setminus B_F)=1-d(B_F)$, which tends to zero as finite truncations exhaust the family. This proves the desired approximation in this case.

Suppose $\beta<1$. For $N\ge2$, let $U_N$ be the union of those full classes whose moduli have no prime factor at most $N$, and let $B_N$ be their actual delayed union. On the product of independent prime-adic Haar coordinates, $U_N$ depends only on primes greater than $N$. The sets $U_N$ decrease, and their intersection is a prime-coordinate tail event. Kolmogorov's zero--one law and $\mu(\bigcap_N U_N)\le\beta<1$ give

$$\mu_N:=\mu(U_N)\longrightarrow0. \tag{3}$$

Here the zero--one law follows because the tail event is independent of every finite set of prime coordinates, hence of the whole product sigma-field and therefore of itself. For each fixed $X$, only finitely many of these classes can be active: their moduli are at most $X$, and there are finitely many residues per modulus. Applying (1) to those classes gives the uniform bound

$$|B_N\cap[1,X]|\le2^r\mu_N X. \tag{4}$$

It remains to handle the nonrough moduli, for a fixed $N$. Partition their classes according to a chosen prime divisor $p\le N$ and the residue $b=a_i\bmod p$, where $0\le b<p$. There are finitely many such groups. In one group write

$$q_i=pm_i,\qquad a_i=b+pc_i.$$

The substitution $n=b+pt$ identifies its delayed exclusions exactly with

$$t\ge m_i,\qquad t\equiv c_i\pmod{m_i}. \tag{5}$$

Indeed $b+pt\ge pm_i$ is equivalent to $t\ge m_i$, because $0\le b<p$. Moreover $\Omega(m_i)\le r-1$. By induction the quotient family has finite-periodic natural approximation. Its image under $t\mapsto b+pt$ has the same property, with densities divided by $p$. Finite unions of these groups retain that property: the omitted union is bounded by the sum of the omitted upper densities.

Given $\varepsilon>0$, first choose $N$ so that $2^r\mu_N<\varepsilon/2$. Then choose finitely many classes in each nonrough group so that the sum of their omitted upper densities is less than $\varepsilon/2$. The union $F$ of these finite selections satisfies

$$\overline d(B\setminus B_F)<\varepsilon$$

by (4). An original finite initial segment can include $F$. Thus $B$ has natural density $\beta$, and its complement has natural density $1-\beta=\delta$. Natural density implies logarithmic density by partial summation. $\square$

Coherent bounded distinct-prime rank

A further proved case allows unbounded exponents. Suppose the residues come from one $\alpha\in\widehat{\mathbb Z}$ and $\omega(q_i)\le r$, where $\omega$ counts distinct prime factors. Then the same natural-density conclusion holds.

For an integer $E\ge2$, a modulus divisible by $p^E$ has its full class contained in $\alpha\pmod{p^E}$. Its delayed hits have $n\ge q_i\ge p^E$. The union over moduli with any exponent at least $E$ therefore has, uniformly in $X$, at most

$$X\sum_p p^{-E}$$

hits below $X$. This tends to zero after division by $X$ as $E\to\infty$. All remaining moduli have $\Omega(q_i)\le r(E-1)$ and are covered by the theorem. Finite-periodic approximation follows by truncating $E$ and then using that theorem. Coherence is essential to this particular high-power bound: without it, the projected residues modulo $p^E$ need not be the same.

Bounded pairwise common rank

Two further natural-density cases follow by a summable-tail reduction. For any fixed $k\ge0$:

• Arbitrary residues are allowed if $\Omega((q_i,q_j))\le k$ whenever $i\ne j$.
• Coherent residues are allowed if $\omega((q_i,q_j))\le k$ whenever $i\ne j$.

For the first assertion, retain the moduli with $\Omega(q)\le k+1$; the established bounded-$\Omega$ theorem applies. For each remaining $q$, let $d$ be the product of its $k+1$ smallest prime factors, counting multiplicity, and let $p$ be the largest prime factor of $d$. No two moduli can have the same $d$, since then their gcd would have at least $k+1$ prime factors counted with multiplicity. Also $q\ge dp$. Consequently

$$\sum_{\Omega(q)\ge k+2}\frac1q \le\sum_{p_1\le\cdots\le p_{k+1}} \frac1{p_1\cdots p_k p_{k+1}^2} \le\sum_{m\ge2}\frac{H_m^k}{m^2}<\infty.$$

The removed family is therefore covered by the summable case.

For the second assertion use the $k+1$ smallest distinct prime factors of every modulus with $\omega(q)\ge k+2$. The same injection and convergent majorant apply. The retained coherent family has bounded $\omega$ and is covered by the preceding theorem. In both cases the two subfamilies have finite-periodic natural approximation, so their union does too.

This reduction does not extend the theorem to arbitrary residues with bounded $\omega(q)$. Indeed the fixed-rank product obstruction shows that even at $\omega(q)\le3$, distinctness and all divisor-face capacities alone cannot supply the needed uniform bound. Its abstract weights have not been realized as first-kill masses.