Structured residues

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The following are proved special cases. They do not cover arbitrary moving residues. Throughout, $\delta$ is the decreasing limit of the densities of the original finite delayed survivor systems.

We use the multiples theorem and finite-periodic logarithmic approximation: for $V\subset\mathbb N$ and finite $V_0\uparrow V$,

$$\operatorname{ld}(M(V)\setminus M(V_0)) =\operatorname{ld}(M(V))-d(M(V_0))\longrightarrow0. \tag{1}$$

The relevant Davenport–Erdős 1951 source passage, equations (2), (4), and (8)–(12), was checked. Alternatively, (1) follows from the self-contained association proof in basic reductions.

Finite rational slopes with integer translations

Theorem. Suppose each original congruence belongs to one of finitely many fixed patterns

$$n_i=dq_i,\qquad a_i\equiv b+cq_i\pmod{dq_i},$$

where the triples $(b,c,d)$ are fixed integers with $d\ge1$ and $0\le c<d$. Then $\operatorname{ld}(A)=\delta$. In particular, this holds when the normalized ratios $a_i/n_i$ belong to a fixed finite subset of $\mathbb Q\cap[0,1)$.

Proof. Fix a triple. Write uniquely $q=sv$, where every prime factor of $s$ divides $d$, and $(v,d)=1$. Partition further by $s$ and $r=v\pmod d$. Let $V$ be the set of $v$ in one such group, and define its actual delayed union

$$T_V=\bigcup_{v\in V} \{n\ge dsv:n\equiv b+csv\pmod{dsv}\}.$$

Apart from the finitely many $n\le b$, its full congruence union is

$$F_V=\{b+sz:z\in M(V),\ z\equiv cr\pmod d\}.$$

Indeed, for $v\mid z$ with $v\equiv r\pmod d$, the congruence $z\equiv cr\pmod d$ is equivalent to $z/v\equiv c\pmod d$. For finite $V_0\subset V$, all its original congruences are active once $n>d s\max V_0$. Therefore, up to finitely many exceptions,

$$T_V\setminus T_{V_0} \subseteq\{b+sz:z\in M(V)\setminus M(V_0)\}.$$

For fixed $b,s$, summing the absolutely summable difference $1/(b+sz)-1/(sz)=O_{b,s}(z^{-2})$ shows that a positive affine image of a set with logarithmic density $\rho$ has logarithmic density $\rho/s$. Thus (1) makes the last tail arbitrarily small in upper logarithmic density.

There are infinitely many possible $s$, but their omitted groups have a uniform natural-density bound. Each deleted integer in an $s$-group satisfies $n\equiv b\pmod s$ and $n\ge dsv\ge s$. Consequently

$$\left|[1,X]\cap\bigcup_{s>Y}T_s\right| \le 2X\sum_{\substack{s>Y\\p\mid s\Rightarrow p\mid d}}\frac1s.$$

The series over these smooth $s$ converges, with total $\prod_{p\mid d}(1-1/p)^{-1}$. First truncate $s$, then finitely approximate each remaining $(s,r)$-group, and finally take the finite union over the original triples. The omitted deletion set then has arbitrarily small upper logarithmic density. An original initial segment contains every chosen finite collection, so

$$\lim_{k\to\infty} \overline{\operatorname{ld}}(A^{(k)}\setminus A)=0.$$

Finite periodic recovery proves the theorem. $\square$

Finite fixed rational translates

Theorem. Suppose finitely many fixed pairs $(b_j,c_j)\in\mathbb Z\times \mathbb N$ are given, and each modulus $n_i$ has an assigned pair satisfying

$$(c_j,n_i)=1,\qquad c_j a_i\equiv b_j\pmod{n_i}.$$

Then $\operatorname{ld}(A)=\delta$.

Proof. First, a primitive set $P$ has logarithmic density zero: its tail is contained in $M(P)\setminus M(P\cap[1,K])$, whose logarithmic density tends to zero by (1).

Fix one pair $(b,c)$ and its modulus set $D$. Put

$$C=\{n\ge1:cn-b>0,\ cn-b\in M(D)\}.$$

For a finite initial part $D_K$, define $C_K$ likewise. Substitution $t=cn-b$ gives

$$\overline{\operatorname{ld}}(C\setminus C_K) \le c\,\operatorname{ld}(M(D)\setminus M(D_K))\longrightarrow0.$$

Here one can use $c/(t+b)=c/t+O_{b,c}(t^{-2})$ and bound the sum over its single congruence class of $t$ by the unrestricted positive sum. Hence $C$ has finite-periodic logarithmic approximation.

Let $P$ be the minimal generators of $D$ under divisibility. Then $P$ is primitive and $M(P)=M(D)$. If $n\in C$ is not deleted by any active modulus, choose $p\in P$ dividing $cn-b$. Necessarily $p>n$. Writing $cn-b=\ell p$ gives $1\le\ell<c+|b|/n$, so all sufficiently large such exceptions belong to a finite union of affine images

$$\{(\ell p+b)/c:p\in P\}\cap\mathbb N.$$

Each has logarithmic density zero. The actual delayed union is contained in $C$ outside the finitely many $n$ with $cn-b\le0$, and the reverse inclusion fails only on the null exceptional set just bounded. Finite unions preserve both conclusions. Finite truncations have the same eventual periodic densities as the corresponding $C_K$, so the limiting survivor density is $\delta$. $\square$

Rational congruence patterns without coprimality

Theorem. Fix finitely many pairs $(C_j,v_j)\in\mathbb Z\times\mathbb N$. Suppose each modulus $q_i$ is assigned a pair satisfying

$$v_j a_i\equiv C_j\pmod{q_i}. \tag{2}$$

No assumption on $(v_j,q_i)$ is needed. The delayed union has finite-periodic approximation in upper logarithmic density, and the survivor set has logarithmic density $\delta$.

This strengthens the preceding rational-translate theorem and subsumes the finite rational-slope theorem. It results from checking the affine endpoint argument in Section 8 of the independent input.

Proof. Fix $(C,v)$. Write $q=hr$, where every prime factor of $h$ divides $v$ and $(r,v)=1$. Partition further by the normalized residue $t=a_q\bmod h$, $0\le t<h$. Necessarily $vt\equiv C\pmod h$, so for each $h$ there are at most $(v,h)\le v$ possible $t$.

For fixed $(h,t)$, write $a_q=t+hA_q$, with $0\le A_q<r$. The substitution $n=t+hm$ identifies the exclusions exactly with

$$m\ge r,\qquad m\equiv A_q\pmod r, \qquad vA_q\equiv (C-vt)/h\pmod r.$$

Activation is exact because $0\le t<h$. The quotient moduli $r$ are distinct in this group, and are coprime to $v$. Thus the preceding rational-translate theorem gives finite-periodic logarithmic approximation for the group. Its positive affine image retains that property.

The groups with $h>Y$ have a uniform counting bound. Each deleted integer in such a group satisfies $n\ge q\ge h$ and $n\equiv t\pmod h$, so there are at most $X/h$ hits below $X$ for each possible $t$. Hence

$$|B_{h>Y}\cap[1,X]| \le vX\sum_{\substack{h>Y\\p\mid h\Rightarrow p\mid v}}\frac1h.$$

The sum tends to zero, being a tail of the finite Euler product $\prod_{p\mid v}(1-1/p)^{-1}$. First truncate $h$, then finitely approximate the remaining groups. A finite union over the pairs $(C_j,v_j)$ proves the assertion. $\square$

Consequence for affine endpoints

Suppose $q_i$ increase, $q_i\le b_i<2q_i$, and $b_j\not\equiv b_i\pmod{q_i}$ for $j>i$. If the pairs $(q_i,b_i)$ lie in finitely many fixed rational affine relations

$$vb_i=uq_i+c\qquad(u,v\in\mathbb N,\ c\in\mathbb Z),$$

then $E=\{b_i\}$ has logarithmic density zero.

For one relation put $a_i=b_i-q_i$. Then $va_i\equiv c\pmod{q_i}$, so (2) applies to its delayed union $B$. Every endpoint is in $B$, whereas every fixed finite prefix $B_F$ meets $E$ in only finitely many points by the avoidance condition. Finite-periodic approximation gives

$$\overline{\operatorname{ld}}(E) \le\overline{\operatorname{ld}}(B\setminus B_F)\longrightarrow0.$$

Take the finite union of the relation-subsequences. This verifies the independent note's special case, including bounded choices of $b_i-q_i$ or $2q_i-b_i$, but not arbitrary or approximately affine endpoints.

Stability under multipliers congruent to one

The abstract lemma in Section 8.6 of the second independent record is valid. Its application to bounded residue orders is already covered by (2), but the following formulation also permits nonperiodic sets.

Theorem. Fix $M\ge1$ and put $S_M=\{k\ge1:k\equiv1\pmod M\}$. Suppose $E\subseteq\mathbb N$ satisfies

$$n\in E,\ k\in S_M\quad\Longrightarrow\quad nk\in E. \tag{3}$$

Then $E$ can be approximated from within, in upper logarithmic density, by finite unions of rays $nS_M$ with $n\in E$. In particular $E$ has logarithmic density. No natural-density conclusion is asserted.

Proof. Let $\mathcal D_M$ consist of the positive integers supported on primes dividing $M$, including $1$. Write each integer uniquely as $du$, with $d\in\mathcal D_M$ and $(u,M)=1$. For a fixed $d$ and a unit residue $r\pmod M$, put

$$T_{d,r}=\{u\ge1:u\equiv r\pmod M,\ du\in E\}, \qquad W_{d,r}=M(T_{d,r}).$$

Then

$$T_{d,r}=W_{d,r}\cap[r]_M. \tag{4}$$

Indeed, if $u=vk$, with $v\in T_{d,r}$ and $u\equiv r\pmod M$, invertibility of $r$ gives $k\equiv1\pmod M$, so (3) applies. The reverse inclusion in (4) is the only nontrivial one.

By the multiples approximation (1), choose finite $G\subset T_{d,r}$ so that $W_{d,r}\setminus M(G)$ has arbitrarily small logarithmic density. Intersecting with $[r]_M$ can only decrease its harmonic mass, and

$$M(G)\cap[r]_M=\bigcup_{v\in G}vS_M.$$

After dilation by $d$, this gives the required approximation in one stratum: dilation divides upper logarithmic density by $d$.

The unretained strata with $d>Y$ have harmonic mass at most

$$H_{\lfloor X\rfloor} \sum_{\substack{d\in\mathcal D_M\\d>Y}}\frac1d.$$

This tail tends to zero after normalization, since

$$\sum_{d\in\mathcal D_M}\frac1d =\prod_{p\mid M}(1-1/p)^{-1}<\infty.$$

There are finitely many remaining $(d,r)$-strata. Approximate each by finitely many rays, then take their union. Every ray is eventually periodic, so these approximations squeeze the upper and lower logarithmic densities of $E$ to the same value. For $M=1$, take the single stratum $d=1$; this recovers the multiples theorem. $\square$

Congruence consequence. Suppose $q_i\mid Ma_i$ for a fixed $M$. More generally than in Problem 25, allow repeated moduli and any finite activation thresholds $T_i$, and set

$$B_i=\{n\ge1:n\ge T_i,\ n\equiv a_i\pmod{q_i}\}.$$

Each $B_i$ satisfies (3): multiplication by $k\equiv1\pmod M$ preserves its congruence and cannot move a positive integer below its threshold. Hence $B=\bigcup_iB_i$ has the ray approximations above. For each selected ray $nS_M$, choose one original $B_i$ containing $n$. That entire ray lies in $B_i$, so any finite ray approximation is contained in a finite original prefix. The prefixes therefore approximate $B$ in upper logarithmic density. Its density equals their limiting periodic density.

For the original activation, this is not a new positive regime: $q_i\mid Ma_i$ is (2) with $(C,v)=(0,M)$, or equivalently $a_i/q_i\in\{0,1/M,\ldots,(M-1)/M\}$ for normalized residues. Likewise Section 8.5's bounded representatives are already covered by the finitely many pairs $(C,v)=(a,1)$. The multiplier theorem supplies an alternative structural proof, not a resolution for arbitrary residues.

Coherent profinite residues: only a partial reduction

If all residues come from one $\alpha\in\widehat{\mathbb Z}$, any modulus that is a multiple of an earlier modulus is redundant, even with delayed activation. Thus the modulus set may be replaced by its primitive core. Its logarithmic density is zero by (1). For each fixed $K$, the union of its first $K$ activated hits per modulus has harmonic mass at most $H_K\sum_{q\le X,\ q\in P}1/q=o(\log X)$. This controls bounded quotients only; it does not prove the full coherent case.

The later coherent power-cutoff estimate controls conditioned tails with $q\le n^{1-\eta}$ for every fixed $\eta>0$. It proves natural density under power-delayed activation, and under the original activation when the primitive core has counting function $O(t/\log t)$. Unrestricted coherent residues remain unresolved; the possible excess is confined to unbounded but subpower quotients.

One cannot simply remove activation for a general profinite $\alpha$. Choose increasing primes $p_m>m$ with $\sum_m1/p_m<\varepsilon$, and prescribe $\alpha\equiv m\pmod{p_m}$ by CRT. The fully active integer trace contains every positive integer. The delayed union instead has natural density $1-\prod_m(1-1/p_m)\le\varepsilon$, by the summable case.