Bounded prime overlap blocks

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The later branch-separated centers theorem allows arbitrarily many blocks at a prime when their centers have bounded multiplicity in each first-digit branch. It proves logarithmic density for arbitrary residues satisfying $\min(a_q,q-a_q)<C P^-(q)$, without coherence, and allows signed centers.

The theorem below extends the disjoint-prime-support result in prepared blocks. It is a sufficient condition, not a resolution for arbitrary singleton residues.

Theorem

Partition a family of globally distinct moduli into nonempty blocks $D_j$. In block $j$, use a single integer residue

$$0\le a_j<\min D_j.$$

Suppose each prime divides moduli in at most $R$ different blocks, where $R$ is a fixed positive integer. Then the delayed union $B$ has logarithmic density equal to the Haar measure of its full periodic union. Moreover, finite congruence subfamilies approximate $B$ in upper logarithmic density.

For a finite such family, if its full union has measure $\beta<1$, then, for every $X\ge1$,

$$\frac{H_B(X)}{H_{\lfloor X\rfloor}} \le \min\left\{1,R\log\frac1{1-\beta}\right\}. \tag{1}$$

No assertion of natural density is made when the full survivor measure is positive.

Product-space inequality

Let $f_1,\ldots,f_m$ be nonnegative functions on a finite product probability space. Suppose each coordinate occurs in the dependence sets of at most $R$ functions. Then

$$\mathbb E\prod_{j=1}^m f_j \le\prod_{j=1}^m(\mathbb E f_j^R)^{1/R}. \tag{2}$$

Here the individual coordinate spaces need not be finite.

For a proof, induct on the number of coordinates. In the last coordinate, apply Hölder with exponent $R$ to its incident factors, padding with factors equal to one if fewer than $R$ occur. Replace each incident $f_j$ by $g_j=(\mathbb E_{\rm last}f_j^R)^{1/R}$, and leave the other factors unchanged. The remaining coordinate-dependence bound is still $R$. The induction hypothesis applies to the $g_j$, and $\mathbb E g_j^R=\mathbb E f_j^R$. This proves (2).

Proof of the theorem

Write

$$U_j=\bigcup_{q\in D_j}[a_j]_q, \qquad \beta_j=\mu(U_j), \qquad \delta=1-\mu\left(\bigcup_j U_j\right).$$

The prime coordinates of $\widehat{\mathbb Z}=\prod_p\mathbb Z_p$ are independent under Haar measure. Apply (2) to the avoidance indicators of finitely many finite block truncations. Exhausting each of those blocks gives, for every finite set $J$ of block indices,

$$\delta\le\mu\left(\bigcap_{j\in J}U_j^c\right) \le\prod_{j\in J}(1-\beta_j)^{1/R}. \tag{3}$$

The passage to infinite blocks is continuity of probability for decreasing sequences of avoidance events.

If $\delta=0$, fixed finite congruence prefixes already prove natural density zero for the survivor set, so assume $\delta>0$. Taking logarithms in (3) and exhausting $J$ yields

$$\sum_j\beta_j \le\sum_j-\log(1-\beta_j) \le R\log(1/\delta)<\infty. \tag{4}$$

Let $B_j$ be the delayed integer union of block $j$. The normalization of its center gives the exact identity

$$B_j=a_j+M(D_j),\qquad M(D_j)=\bigcup_{q\in D_j}q\mathbb N. \tag{5}$$

Indeed a congruence hit is active precisely at the points $a_j+tq$ with $t\ge1$; its root $a_j<q$ is inactive. Consequently, using the exact harmonic inequality for multiples,

$$H_{B_j}(X) \le H_{M(D_j)}(X) \le\beta_j H_{\lfloor X\rfloor}. \tag{6}$$

For infinite $D_j$, at a fixed cutoff only the finitely many moduli $q\le X$ matter, so the finite inequality applies directly.

Equations (4)--(6) give the uniform tail bound

$$H_{\bigcup_{j>J}B_j}(X) \le H_{\lfloor X\rfloor}\sum_{j>J}\beta_j. \tag{7}$$

Each retained block has finite-congruence approximation in upper logarithmic density: this is the multiples theorem followed by the fixed translation in (5). More explicitly, for a finite subfamily $F\subset D_j$, the multiples difference has logarithmic density $\beta_j-\mu(M(F))$, which tends to zero as $F$ exhausts $D_j$. Fixed translations preserve logarithmic density, because their reciprocal-weight changes are absolutely summable. Finite unions of retained blocks therefore have the same approximation property. Combining it with (7) proves the theorem. The limit is the full Haar measure, since the finite periodic union measures increase to it.

For a finite family, summing (6) and using (4), with $\delta=1-\beta$, proves (1). Global modulus distinctness is preserved throughout: blocks are a partition of the given modulus set, not additional copies of congruences.

A coherent varying-residue consequence

Suppose $a_q=\alpha\bmod q$, normalized by $0\le a_q<q$, for a fixed $\alpha\in\widehat{\mathbb Z}$. Omit the trivial modulus-$1$ case. If, for some fixed $C>0$,

$$a_q<C P^-(q)\qquad\text{for every selected }q, \tag{8}$$

where $P^-(q)$ is the least prime factor, then the original delayed system has logarithmic density and finite-periodic logarithmic approximation.

To prove this, group the moduli by their exact normalized residue $a_q=a$. If a prime $p$ occurs in a group, then

$$a\equiv\alpha\pmod p,\qquad 0\le a<Cp.$$

At most $\lceil C\rceil$ integers in this interval have that residue modulo $p$. Hence the block theorem applies with $R=\lceil C\rceil$.

Condition (8) may also hold only outside a finite set of moduli: the exceptional moduli introduce at most finitely many additional center groups and thus increase $R$ by at most a fixed amount.

For an unrestricted coherent center, a prime can occur in arbitrarily many normalized-residue groups. Neither (3) nor (4) then supplies the summable block bound. This is the remaining limitation of this argument.

Allowing a reciprocal-summable set of exceptional overlap primes

Theorem. Suppose the residues are coherent, and partition the moduli by their exact normalized residues $a_q=a$. Let $\mathcal P$ be a set of primes with $\sum_{p\in\mathcal P}1/p<\infty$. If each prime outside $\mathcal P$ occurs in at most $R$ of these groups, then the delayed union has logarithmic density and finite-congruence approximation in upper logarithmic density.

There is no bound on how many groups share a prime in $\mathcal P$. In particular, arbitrary overlap at finitely many primes is harmless. The proof does not give a uniform finite small-footprint estimate when the exceptional set and the needed truncations vary.

Finite exceptional set

First suppose $\mathcal P$ is finite and nonempty. Fix $K\ge1$, and temporarily discard every modulus divisible by $p^K$ for any $p\in\mathcal P$. Coherence and activation imply that all discarded hits lie in the union of the delayed classes $[\alpha]_{p^K}$. Thus their harmonic mass is at most

$$H_{\lfloor X\rfloor}\sum_{p\in\mathcal P}p^{-K}. \tag{9}$$

Their full Haar union has at most the same sum as its measure.

Put $L=\prod_{p\in\mathcal P}p^K$ and consider the retained family. In a fixed fiber $n=r+Lm$, $0\le r<L$, split each root group $a$ further by $h=(q,L)$. Every retained modulus in this subblock has the form $q=hd$, where $(d,L)=1$. Write $s=L/h$. The subblock is incompatible with the fiber unless $a\equiv r\pmod h$. For a compatible subblock, put $b=(a-r)/h$. For every one of its moduli, normalization gives

$$-s<b<d.$$

The original active hits, with $m\ge1$, are exactly

sm-b\in M(D'),\qquad D'=\{q/h:q\text{ is in the subblock}}. \tag{10}

Indeed $n-a=h(sm-b)$; being a positive multiple of $q=hd$ is equivalent to (10). It automatically gives $n\ge a+q\ge q$. For such a hit, $k=sm-b>0$ and

$$k<s(m+1)\le2sm, \qquad \frac1m\le\frac{2s}{k}.$$

Since $m\mapsto k$ is injective, the exact multiples inequality gives

$$H_{\text{projected subblock}}(T) \le2s\,\beta_{D'} H_{\lfloor s(T+1)\rfloor}, \qquad \beta_{D'}=\mu(M(D')). \tag{11}$$

The full projected union has measure exactly $\beta_{D'}$: multiplication by $s$ and translation by $-b$ preserve Haar measure on all coordinates appearing in $D'$, since $(d,s)=1$.

Outside $\mathcal P$, a prime now occurs in at most $R\tau(L)$ projected subblocks, where $\tau(L)$ is the number of divisors of $L$. No prime in $\mathcal P$ occurs in $D'$. Let $\delta_r$ be the full survivor measure of the projected family in the chosen fiber. If $\delta_r>0$, product-space Hölder, as in (3)--(4), gives

$$\sum_{\text{compatible subblocks}}\beta_{D'} \le R\tau(L)\log(1/\delta_r)<\infty. \tag{12}$$

Here $L$ is fixed. The ratios $H_{\lfloor s(T+1)\rfloor}/H_{\lfloor T\rfloor}$ are bounded by a constant depending only on $L$, uniformly for $T\ge1$. Consequently (11)--(12) give uniformly negligible harmonic tails of subblocks.

Each individual subblock also has finite-congruence logarithmic approximation. Under (10), deleting all but finitely many generators restricts a difference of two multiples sets to one residue modulo $s$. The unrestricted multiples difference has logarithmic density tending to zero as the generators are exhausted; the inequality $1/m\le2s/k$ transfers its upper harmonic bound. This suffices even though no density for the restricted multiples difference is being assumed. Thus (12) yields finite-congruence approximation for the entire fiber.

If $\delta_r=0$, finite prefixes of the projected family have survivor densities tending to zero. Each such prefix is eventually periodic, so its complement bounds the tail and gives the same approximation conclusion. The finitely many original integers represented by $m=0$ do not matter. Finally, the affine map $m\mapsto r+Lm$ divides upper logarithmic density by $L$. Summing over the finitely many fibers establishes the conclusion for the retained family.

Let $K\to\infty$ in (9). Uniform harmonic approximation and the corresponding Haar bound transfer finite-congruence approximation and the Haar value to the original family. If $\mathcal P$ is empty, the original bounded-overlap theorem applies directly.

Infinite reciprocal-summable exceptional set

First discard moduli containing any $p\in\mathcal P$ with $p>Y$. Every discarded active hit is in the delayed class $[\alpha]_p$ for such a prime, so its normalized harmonic mass, and the Haar measure of the discarded union, are at most

$$\sum_{\substack{p\in\mathcal P\\p>Y}}\frac1p.$$

For the retained family, the only exceptional primes are the finitely many in $\mathcal P\cap[1,Y]$. Apply the finite-exception argument and then let $Y\to\infty$.

All subblocks are partitions or restrictions of the original distinct moduli. Effective moduli may repeat across subblocks, but Hölder is applied to subblock events, not to a newly asserted singleton system. Thus no duplicate-modulus construction is introduced. Coherence is essential for both exceptional-prime deletion bounds. The argument does not handle unbounded overlap on a set of primes with divergent reciprocal sum.