Dyadic splitting

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The statements below are proved reductions, not a counterexample. They show that an infinite, measure-preserving binary refinement does not bypass the positive-private-witness problem in unresolved estimates.

Exact delayed lift

Let the base moduli $q$ be distinct and odd, with normalized residues $0\le a_q<q$. Replace the base class by

$$a_q+2^{k-1}q\pmod{2^kq},\qquad k\ge1. \tag{1}$$

All the new moduli are globally distinct: their odd part identifies $q$ and their 2-adic valuation identifies $k$. The numerator displayed in (1) is already between zero and $2^kq$.

Inside the base cylinder, put $n=a_q+qt$. The full cylinders (1) partition the quotient values with finite 2-adic valuation. Their omitted 2-adic fiber has Haar measure zero. Thus the countable full lifted union has the same Haar measure as the full base union, also when base cylinders overlap.

For a positive integer $t=2^v u$, $u$ odd, its unique lifted class has modulus $2^{v+1}q$. It is active exactly when

$$a_q+qt\ge 2^{v+1}q,$$

equivalently $t\ge2^{v+1}$, because $0\le a_q/q<1$. This is equivalent to $u\ge3$. Consequently the delayed lift misses precisely the power-of-two quotients of each individual base class. The lift is a subset of the delayed base union.

Write $L=B_{\rm base}\setminus B_{\rm lift}$. If $n\in L$, then for every active base class containing $n$, its quotient is a power of two. This qualification is essential when the base classes overlap.

Positive loss gives positive private endpoints

Let $\beta$ be the full base-union Haar measure. The established compression bound gives

$$\sum_{q\le X}\frac1q\le\beta H_{\lfloor X\rfloor}. \tag{2}$$

Thus points in $L$ whose chosen witnessing quotient is $2^v$ with $v\ge R$ have harmonic mass at most

$$2^{1-R}\beta H_{\lfloor X\rfloor}. \tag{3}$$

If $L$ has positive upper logarithmic density, some fixed $v$ therefore gives a positive-upper-logarithmic-density set of points $n=a_q+2^vq$. Restrict further to one residue modulo $2^{v+1}$, keeping positive upper logarithmic density. Give each selected point modulus $Q_q=2^vq$. These moduli are distinct and $Q_q\le n<2Q_q$.

Each cylinder $[n]_{Q_q}$ contains only its own selected point. Indeed, another selected point $m$ in this cylinder has the form $m=a_q+2^wq$ if $m\ge q$, since it lies in $L$. Congruence modulo $2^vq$ implies $w\ge v$; if $w>v$, then $m-n$ has exact 2-adic valuation $v$, contradicting the restriction modulo $2^{v+1}$. If $w=v$, then $m=n$. The only possible point of the base cylinder below $q$ is $a_q$, and it also differs from $n$ by an integer of exact 2-adic valuation $v$. Thus the selected witnesses are globally private.

Conversely, any positive-upper-logarithmic-density private family $q\le r_q<2q$ can be reduced to one with odd moduli. The witnesses whose moduli are divisible by $2^R$ have harmonic mass at most

$$\sum_{\substack{q\le X\\2^R\mid q}}\frac1q \le2^{-R}H_{\lfloor X\rfloor}.$$

Choose a fixed valuation $v_2(q)=v$, and then a fixed witness residue $b\pmod{2^v}$, retaining positive upper logarithmic density. Write $q=2^vu$, $r_q=2^vs_u+b$. The moduli $u$ are odd and distinct, $u\le s_u<2u$, and privacy is preserved. Affine rescaling preserves positivity of upper logarithmic density.

For this odd family set $a_u=s_u-u$. Its private witnesses all have quotient one. No witness belongs to another base cylinder, and the lift (1) misses its own quotient-one witness. Hence its loss contains the entire private witness set. This proves the equivalence:

Every binary lift (1) has logarithmically null loss if and only if every distinct-modulus private family $q\le r_q<2q$ has logarithmic density zero.

The endpoint assertion remains unproved.

Arbitrary 2-adic splitting centers give the same obstruction

One may instead omit any chosen quotient fiber $c_q\in\mathbb Z_2$. At level $k$, use the unique residue modulo $2^k$ agreeing with $c_q$ in the first $k-1$ bits and disagreeing in the next bit. As before, the full lifted union agrees with the base cylinder up to a null set, and all lifted moduli are distinct.

The positive quotients missed by delayed activation are exactly

$$T(c_q)=\{2^j+(c_q\bmod2^j):j\ge0\}. \tag{4}$$

To check this, let $2^j\le t<2^{j+1}$. The first disagreeing bit gives an inactive class precisely when $t\equiv c_q\pmod{2^j}$; equality $t=c_q\in\mathbb Z_2$ is omitted altogether and satisfies the same condition. There is exactly one such $t$ in this dyadic interval. In particular

$$\sum_{t\in T(c_q)}\frac1t\le2, \qquad \sum_{\substack{t\in T(c_q)\\t\ge2^R}}\frac1t\le2^{1-R}. \tag{5}$$

Equations (2) and (5) show that positive upper logarithmic loss supplies a positive-mass set $E$ with one selected point $r_q$ per modulus and $q\le r_q\le Kq$, for some fixed $K$. Greedily retain these points in increasing modulus order, rejecting a point if it lies in an earlier retained cylinder. Charge a retained point also to itself. A retained point $r_q$ receives total harmonic mass at most $2/q\le2K/r_q$, by (5): every later rejected endpoint is at least $q$, so no inactive root is involved. For every $X$, all endpoints of $E\cap[1,X]$ are charged to retained points at most $KX$. Hence

$$H_E(X)\le2K H_{E'}(KX).$$

The retained first-kill endpoints $E'$ have positive upper logarithmic density. The finite coloring and quotient rescaling already proved in the private-witness reduction give a positive-density private family in $[q,2q)$. Thus nonzero splitting centers do not evade the same endpoint problem.

Why a positive private family would settle the original problem negatively

If $E=\{r_q\}$ were such a private family with upper logarithmic density $\alpha>0$, let $B$ be its own delayed congruence union and $\beta$ its limiting periodic density. For every finite prefix $B_F$, its intersection with $E$ is finite. Since $B\supseteq B_F\cup E$,

$$\overline{\operatorname{ld}}(B)\ge d(B_F)+\alpha.$$

Letting the prefix grow gives $\overline{\operatorname{ld}}(B)\ge \beta+\alpha$. The cutoff-preserving subfamily extraction in basic reductions then gives a system whose logarithmic density does not exist. No positive-density private family has been constructed here.