Spectral sieve

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This approach studies Fourier mass rather than the size of individual quotient-sieve spikes. The identities below are proved. They do not yet give the required estimate along the activated integer orbit.

Let $G=\widehat{\mathbb Z}$ with Haar probability $\mu$. A rational character $\chi\in\mathbb Q/\mathbb Z$ has conductor $d(\chi)$, its order. Fourier transforms use $\widehat f(\chi)=\int f(x)\overline{\chi(x)}\,d\mu(x)$. For a finite family put

$$C_i=[a_i]_{q_i},\quad U=\bigcup_i C_i,\quad F_i=C_i\setminus\bigcup_{j<i}C_j,\quad e_i=\mu(F_i).$$

Logarithmic conductor energy

For every finite family, even allowing repeated moduli,

$$\sum_\chi \log d(\chi)\, |\widehat{\mathbf1_U}(\chi)|^2 \le\sum_i e_i\log q_i. \tag{1}$$

Proof. Work on the finite group modulo $L=\operatorname{lcm}_i q_i$. For $p\mid L$ and $1\le k\le v_p(L)$, let $T_{p,k-1}$ average the $p$-adic digits at levels $k$ and above, fixing all other coordinates and the residue modulo $p^{k-1}$. Its Fourier multiplier is $\mathbf1_{v_p(d(\chi))<k}$.

Couple Haar $x$ with a conditionally independent resampling $y$ in that fiber. For $f=\mathbf1_U$,

$$\|f-T_{p,k-1}f\|_2^2 =\mathbb P(f(x)=1,f(y)=0).$$

If $x\in F_i$ and $v_p(q_i)<k$, the resampling preserves membership in $C_i$, so $f(y)=1$. Therefore

$$\sum_{v_p(d(\chi))\ge k}|\widehat f(\chi)|^2 \le\sum_{p^k\mid q_i}e_i.$$

Multiply by $\log p$ and sum over $(p,k)$ to obtain (1). $\square$

The survivor indicator has the same nonconstant Fourier coefficients up to sign, so satisfies the same bound. For a fixed infinite family, let $U_X$ use the moduli at most $X$. Since $\sum_i e_i\le1$,

$$\sum_{q_i\le X}e_i\log q_i=o(\log X). \tag{2}$$

Indeed, a fixed initial part contributes $O(1)$, and the remaining sum is at most $\log X$ times its arbitrarily small total $e_i$-mass. Thus (1) bounds the full finite-prefix Fourier energy by $o(\log X)$. This is not a pointwise Fourier approximation theorem.

Exact update for a singleton class

Add $C=[a]_q$ to an old union with indicator $f$. Write

$$h=\mathbf1_{C\setminus U},\quad e=\mu(C\setminus U),\quad f'=f+h,$$

and let $P_q$ be conditional expectation onto functions modulo $q$. Then

$$P_qh=qe\,\mathbf1_C, \qquad \|(I-P_q)f'\|_2^2-\|(I-P_q)f\|_2^2=-e+qe^2. \tag{3}$$

To verify the second identity, only the fiber $C$ changes. On that fiber $P_qf=1-qe$, and after the installation it is identically one. Since $\|f'\|_2^2-\|f\|_2^2=e$, whereas

$$\|P_qf'\|_2^2-\|P_qf\|_2^2 =q^{-1}\bigl(1-(1-qe)^2\bigr)=2e-qe^2,$$

subtracting proves (3). Thus the total Fourier mass at conductors not dividing $q$ decreases by $e(1-qe)$.

There is a finer blockwise identity. Put $H_q=\{j/q\bmod1:0\le j<q\}$, and partition the dual group into cosets of $H_q$. On a nonzero coset $\chi+H_q$, set $v_h=\widehat f(\chi+h)$ for $h\in H_q$. Since the Fourier transform of $\mathbf1_C$ is $q^{-1}\exp(-2\pi i a h)$ on $H_q$,

$$v'_h=v_h-\frac{\exp(-2\pi i a h)}q \sum_{k\in H_q}\exp(2\pi i a k)v_k. \tag{4}$$

This is an orthogonal rank-one projection. The squared norm decreases on each nonzero coset separately; only the zero coset receives the additional term from $\mathbf1_C$. Multiple residues at one modulus would impose several orthogonal projections at that same step. Singleton distinctness allows only one installation per $q$.

Why individual-increment truncation is invalid

The checked translated digit sieve has distinct old moduli at most $N=M^2$, with full survivor class $Q\pmod{L_N}$, where

$$Q=N+1,\qquad L_N\ge2^M/(M+1).$$

Appending $[0]_Q$ gives a first-kill cell which contains the active endpoint $Q$ and is one class modulo $R=\operatorname{lcm}(Q,L_N)$. Its Fourier coefficients have magnitude $1/R$ at the $R$ characters whose conductors divide $R$. The number of rational characters with conductor at most $Q^A$ is $O(Q^{2A})$. Hence, for every fixed $A$, its conductor-$\le Q^A$ Fourier truncation has absolute value

$$O(Q^{2A}/R)=o(1)$$

at $Q$, although the cell's indicator equals one there.

Activation and distinctness therefore do not justify a polynomial-height pointwise truncation of individual first-kill increments. Any successful use of (3)--(4) must retain cancellation against the old spectrum. Also, (1) alone cannot distinguish the singleton problem from the multi-residue setting, where logarithmic density can fail. No global summation of these spectral updates giving $o(\log X)$ harmonic discrepancy has been proved.

The proposed energy-charging estimate is false

A proposed use of the first-kill energy was the uniform finite estimate

$$H_B(X)\le\beta H_{\lfloor X\rfloor} +C\left(1+\sum_{q_i\le X}e_i\log q_i\right), \tag{5}$$

with an absolute $C$, for distinct singleton moduli. Here $\beta$ is the full finite union measure. This estimate is false.

The checked energy counterexample gives finite distinct-modulus singleton systems, with the original delayed activation, for which

$$\sum_q e_q\log q\longrightarrow2\log2, \qquad H_B(Q)-\beta H_Q\sim(\log Q)^{1/21}\longrightarrow\infty.$$

The energy includes every preparatory congruence. Thus this is an amortized counterexample to (5), not merely a failure of an individual-cell comparison. It does not contradict (1) or the exact spectral updates (3)--(4).

Its former sufficiency argument remains correct: apply it to the finite prefix $q_i\le X$ of a fixed infinite family. Its deletion set agrees with the infinite deletion set below $X$, its measure is at most $\beta_\infty$, and its first-kill densities agree with the infinite family's corresponding densities. Equation (2) would then give $\limsup H_B(X)/\log X\le\beta_\infty$. Fixed prefixes give the opposite lower bound.

The counterexample's normalized excess tends to zero, so it does not refute a uniform additive $o(\log X)$ estimate or the infinite problem. Any further spectral proof must obtain an orbit comparison other than the false bound (5).

A weaker energy target: unresolved

The continuation considered, but did not prove or refute, the estimate

$$(H_B(X)-\beta H_X)_+ \le C\sqrt{H_X\left(1+\sum_{q_i\le X}e_i\log q_i\right)}. \tag{6}$$

Together with (2), it would prove the exact problem. It is much weaker than the false linear-energy estimate (5). The newer rank-filtered bounded-energy examples have excess at least $(\log X)^\gamma$ for every $\gamma<(7\,2^{1/7}-3\,3^{1/3}-3)/4=0.100469\ldots$. For their parameters the trivial survivor upper bound also gives excess $o(\sqrt{\log X})$. Thus they do not refute (6).

One exact restriction on possible counterexamples to (6) is available. Write $\delta=1-\beta$ and

$$W=\sum_{\substack{p\le X\\p\text{ is a selected prime modulus}}} \frac{\log p}{p}.$$

For a selected prime modulus $p$, every earlier modulus is smaller than $p$, hence coprime to it. Consequently its first-kill measure is $e_p=\delta_{p-}/p\ge\delta/p$. If $\mathcal E=\sum e_q\log q$ and $D_+=(H_B-\beta H_X)_+$, then

$$\mathcal E\ge\delta W,\qquad D_+\le\delta H_X, \qquad \frac{D_+^2}{H_X\mathcal E}\le\frac{\delta H_X}{W} \tag{7}$$

whenever the denominators are positive. Thus (6) holds on families with $W\ge c\delta H_X$, with a constant depending only on $c>0$. This is not a general estimate: arbitrary families can have $W=0$. No unverified prime-distribution theorem is needed or used in (7).

The remaining spectral gap is specific. A bound for arbitrary disjoint periodic functions $h_q\le\mathbf1_{C_q}$ cannot substitute for (6): such functions could select extremely small high-period subcylinders around the sampled integers. The actual recursion $h_q=\mathbf1_{C_q}(1-f_{\rm old})$ must be retained. No summation of its projection identities proving (6) was obtained.

Isolating the negative energy term in the activated trace

There is a checked refinement of the exact projection identity, but still no bound for its remaining trace. Put

$$h_q=\mathbf1_{F_q},\qquad r_q=(I-P_q)h_q, \qquad R_q(X)=\sum_{q\le n\le X}\frac{r_q(n)}n.$$

Here $P_qh_q=qe_q\mathbf1_{C_q}$. With $B_q=C_q\cap[q,\infty)$, the exact identity is

$$H_{E_q}(X)=qe_qH_{B_q}(X)+R_q(X). \tag{8}$$

Since $qH_{B_q}(X)\le H_{\lfloor X/q\rfloor}$, summation gives

$$\begin{aligned} H_B(X)-\beta H_{\lfloor X\rfloor} &\le\sum_{q\le X}R_q(X) -\sum_{q\le X}e_q (H_{\lfloor X\rfloor}-H_{\lfloor X/q\rfloor})\\ &\le\sum_{q\le X}R_q(X)-\mathcal E_X+\log2, \qquad \mathcal E_X=\sum_{q\le X}e_q\log q. \end{aligned} \tag{9}$$

For the last line use $H_N-H_{\lfloor N/q\rfloor}\ge\log(q/2)$ and $\sum e_q\le1$. If the finite family contains inactive moduli above $X$, their contribution to $\beta$ only improves the first inequality.

Orthogonality of conditional expectation also gives

$$\|r_q\|_2^2=e_q-qe_q^2,$$

exactly the decrease of the old spectrum outside the characters whose conductors divide $q$. Thus the low-conductor part of each increment already supplies a negative energy charge in (9). What would suffice is the still-unproved simultaneous bound

$$\sum_{q\le X}R_q(X) \le\mathcal E_X+C\sqrt{H_{\lfloor X\rfloor}(1+\mathcal E_X)}. \tag{10}$$

It would imply (6), up to an absolute additive constant.

Separate Cauchy--Schwarz estimates do not establish (10): the point-evaluation kernels on the common period can have arbitrarily large high-conductor norm. Pairwise progression error estimates also do not control the nonlinear first-kill recursion. The sharp nonempty-cell bound quantifies a singleton restriction on such cells, but does not bound the sum in (10). No global trace estimate resulted from this attempt.

A cumulative quotient-sieve energy budget

There is a stronger energy inequality than (1), retaining the energy of every first-kill innovation rather than only the final union. Put

$$f_i=\mathbf1_{\bigcup_{j\le i}C_j},\quad h_i=f_i-f_{i-1},\quad e_i=\mu(F_i),\quad \mathcal E(f)=\sum_\chi\log d(\chi)|\widehat f(\chi)|^2.$$

For every finite family,

$$\mathcal E(f_N)+ \sum_{i\le N}\sum_\chi \log\frac{d(\chi)}{\gcd(d(\chi),q_i)} |\widehat h_i(\chi)|^2 \le\sum_{i\le N}e_i\log q_i. \tag{11}$$

Proof. Work on a finite common period and fix the prime-digit conditional expectation $T=T_{p,k-1}$, with $Q=I-T$. If $k>v_p(q_i)$, the cylinder $C_i$ is $T$-measurable. Therefore

$$Qh_i=-\mathbf1_{C_i}Qf_{i-1},\qquad Qf_i=(1-\mathbf1_{C_i})Qf_{i-1},$$

and

$$\|Qf_i\|_2^2-\|Qf_{i-1}\|_2^2=-\|Qh_i\|_2^2.$$

If $k\le v_p(q_i)$, orthogonality of conditional expectation and disjointness of $f_{i-1}$ and $h_i$ instead give

$$\|Qf_i\|_2^2-\|Qf_{i-1}\|_2^2 =e_i-2\langle Tf_{i-1},Th_i\rangle-\|Th_i\|_2^2 \le e_i.$$

Both conditional expectations in the inner product are nonnegative. Telescoping proves the coordinatewise inequality

$$\|Qf_N\|_2^2+ \sum_{i:v_p(q_i)<k}\|Qh_i\|_2^2 \le\sum_{i:v_p(q_i)\ge k}e_i.$$

Multiply by $\log p$ and sum over $p,k$. A character in the innovation term occurs at exactly the levels $v_p(q_i)<k\le v_p(d(\chi))$, proving (11). $\square$

The quotient-survivor indicator

$$g_i(t)=1-f_{i-1}(a_i+q_it)$$

has mean $q_ie_i$. Extension from the cylinder $C_i$ and Fourier pullback under $t\mapsto a_i+q_it$ give

$$\sum_\chi\log\frac{d(\chi)}{\gcd(d(\chi),q_i)} |\widehat h_i(\chi)|^2 =\frac1{q_i}\mathcal E(g_i).$$

Indeed, explicitly,

$$\widehat h_i(\chi)=q_i^{-1}e^{-2\pi i a_i\chi} \widehat g_i(q_i\chi).$$

For a common period $L$ divisible by $q_i$, the function $g_i$ has period $L/q_i$. Multiplication by $q_i$ maps the characters modulo $L$ onto those modulo $L/q_i$, with fibers of size $q_i$. The quotient character has conductor $d(\chi)/\gcd(d(\chi),q_i)$. Thus

$$\mathcal E(f_N)+\sum_{i\le N}\frac{\mathcal E(g_i)}{q_i} \le\sum_{i\le N}e_i\log q_i. \tag{12}$$

For a fixed infinite family, (2) bounds the cumulative quotient-sieve energy over $q_i\le X$ by $o(\log X)$.

Neither inequality uses distinctness of the moduli; repeated single-cylinder installations satisfy the same argument. They do not control high-conductor point evaluation on the activated integer orbit, and cannot alone distinguish the singleton problem from its multi-residue variant. The entropy and square-root trace estimates remain unproved.