bibkey: baker2021smoothbeatty authors: Roger Baker year: 2021 title: Smooth numbers in Beatty sequences doi: 10.4064/aa210322-22-6 url: https://arxiv.org/abs/2102.00303v1 claim: “Theorem 1 counts smooth Beatty values above a log-power-three smoothness threshold; complete CA layer support prevents fixed primorial peeling from reaching that range, while a classical Dirichlet application diagnoses weaker sampling conditions.” strata_touched: [] license: citation-only triage: anchor

Smooth Beatty suppliers at the actual CA scale

Roger Baker, Smooth numbers in Beatty sequences, Acta Arithmetica 200 (2021), 429–438, DOI:10.4064/aa210322-22-6. The publisher record and the primary text arXiv:2102.00303v1, §1, Theorem 1 and equation (1.1), were inspected. The full analytic proof was not independently verified. The following applications reuse that statement and classical inputs; they introduce no new smooth-number theorem, originality claim or Lean verification.

The published range and the exact unit-bit population

For fixed finite-type irrational $\theta >1$ and fixed $\beta \ge 0$, set

$${\mathcal{B}}_{\theta ,\beta }(X)=\{\lfloor \theta k+\beta \rfloor :1\le k\le X\}.$$

For every fixed $\epsilon >0$, Theorem 1 gives

$$\#\{m\in {\mathcal{B}}_{\theta ,\beta }(X):P^{+}(m)\le y\}={\theta }^{-1}\Psi (\theta X,y)(1+o(1))$$

as $X\to \infty$, uniformly for $(\log X{)}^{3+\epsilon }\le y\le X$. Here $\Psi (T,y)$ counts positive $y$-smooth integers at most $T$. The theorem fixes the slope; it is a counting asymptotic, not a pointwise bound for an optimizing integer.

The existing canonical FIB unit-bit interface identifies $h(n)=1$ with $n=\lfloor k{\phi }^2-1\rfloor$, $k\ge 1$. Its negative offset must not be inserted directly into Baker’s nonnegative-offset statement. Instead, for $k\ge 2$, put $j=k-1$:

$$\lfloor k{\phi }^2-1\rfloor =\lfloor j{\phi }^2+\phi \rfloor ,j\ge 1.$$

Thus $(\theta ,\beta )=({\phi }^2,\phi )$ gives the same unit-bit population except for its first value, 1. This is an exact parameter map to the published counting theorem. For an actual CA candidate $n\asymp X$, however, the existing support asymptotic is $P^{+}(n)\sim \log n\asymp \log X$. Its smoothness scale is outside the stated range. Enlarging $y$ to meet that range counts a larger population and supplies no restriction back to the actual CA test set.

Peeling complete layers does not repair the smoothness scale

Use the classical CA exponents, including every intermediate tied maximizer, and write $n={\prod }_{p\le P}{p}^{e_p}$ with $P=P^{+}(n)$. For a fixed nonnegative integer $j$, remove the first $j$ complete exponent layers and retain the exact quotient

$$v_j=\prod _{p\le P}{p}^{\max (e_p-j,0)}.$$

By the existing nonincreasing-exponent property, its nonempty support is the complete prime prefix through $y_j=P^{+}(v_j)$. Hence

$$y_j\#\mid v_j,\vartheta (y_j)\le \log v_j,y_j\le (1+o(1))\log v_j$$

when $y_j\to \infty$, using ordinary PNT. For actual CA sequences with $n\to \infty$ and fixed $j$, the classical fixed-layer threshold gives $y_j\to \infty$; see the existing (T11) application in the CA mask note. At a natural new population scale $X_j\asymp v_j$, every fixed $C>1$ therefore has

$$y_j<(\log X_j{)}^C$$

eventually. In particular, removing any fixed number of complete layers still does not enter Baker’s $C=3+\epsilon$ range. No assertion that $v_j$ itself is CA is needed or made.

For one removed layer, $R=P\#$ and $v=n/R$ also give a same-source incidence map. In the actual $h=1$ branch, for an eligible $p\ne 2,5$ with even $e_p>0$, put $d_p=p^{e_p/2}$. Then $d_p\mid v$ and the existing lift $c=2\lfloor n\phi \rfloor -n-1$ gives

$$p^{e_p/2}\mid c⟺\left\{\frac{v}{d_p}(R\phi )\right\}\in I_{d_p},I_d=\left(\frac{d+1+2a}{2d},\frac{d+1+4a}{2d}\right),a=2-\phi .$$

Indeed, $d_p\mid n$ is odd and at least three. Writing $n=d_{p}t$, $x=\{t\phi \}$ and $k=\lfloor d_{p}x\rfloor$ gives $c\equiv 2k-1(\mathrm{mod}{d}_p)$; the unique possible value is $k=(d_p+1)/2$. Intersect with the existing unit window $a<\{n\phi \}<2a$. The empty-product value $d=1$ is excluded, and prime five retains its separate rule. These are the actual same-integer depth tests, without an independence assumption. The new slope $R\phi$ also varies with $P$, so Baker’s fixed-slope statement cannot silently provide its uniform error.

A controlled comparator diagnoses weaker sampling hypotheses

Complete support and the correct size scale alone do not force rotation equidistribution. The classical finite Dirichlet theorem supplies, for every real $\alpha$ and integer $Q\ge 1$, an integer $1\le t\le Q$ with $\Vert t\alpha \Vert \le 1/(Q+1)$; no new approximation proof is needed. A matching upstream statement is Real.exists_nat_abs_mul_sub_round_le in the pinned Mathlib source. It was inspected, not compiled here.

Take a sequence of actual CA integers $C_i\to \infty$, so that $P_i=P^{+}(C_i)\to \infty$. Put $Q_i=\lfloor \log P_i\rfloor$ and discard the finite initial range with $Q_i<1$. Apply the classical input to $\alpha =C_i\phi$. The resulting $N_i=t_i{C}_i$ satisfy

$$\begin{array}{c}\mathrm{rad}(N_i)=\mathrm{rad}(C_i)=P_i\#,P^{+}(N_i)=P_i,\\ \log N_i=\log C_i+O(\log \log P_i)\sim P_i,\\ v_p(N_i)=v_p(C_i)(p>Q_i),\Vert N_i\phi \Vert \le 1/(Q_i+1)⟶0.\end{array}$$

Every prime factor of $t_i\le Q_i<P_i$ already belongs to $C_i$. Thus this enlarged population retains initial support, size scale and all exponents above $\log P_i$, while the chosen phases concentrate at zero on the circle. The $N_i$ are not asserted to be CA or to have nonincreasing exponents. In fact their unit bits are eventually zero: a shrinking circular neighborhood of zero misses the fixed interval $(a,2a)$ for unit bit one. This is a counterexample to the stated weaker sampling premise, not to a hypothesis restricted to $h=1$ CA integers or to Robin.

The comparator also preserves the Robin response to a finer scale than the existing square-root margin. Choose an optimizing CA price ${ϵ}_i$ for $C_i$. Comparison with $C_i/P_i$ bounds it by

$$0<{ϵ}_i\le \frac{\log (1+1/P_i)}{\log P_i}=O\left(\frac{1}{P_i\log P_i}\right):$$

the removed last local factor has ratio at most the first gain $1+1/P_i$. This price bound also holds at tied choices; the existing Nicolas source note records the classical optimizing objective. Monotonicity of each finite Euler factor under divisibility gives $Z(N_i)\ge Z(C_i)$. Optimality and $t_i\le Q_i$ then imply

$$0\le {\mathrm{Ben}}_{C_i,{ϵ}_i}(N_i)\le {ϵ}_i\log t_i,0\le \log \frac{Z(N_i)}{Z(C_i)}\le {ϵ}_i\log t_i.$$

Set $\mathcal{R}(m)=Z(m)/(e^{\gamma }\log \log m)$. Since the derivative of $\log \log L$ is $1/(L\log L)$, the same comparison gives

$$\left|\log \mathcal{R}(N_i)-\log \mathcal{R}(C_i)\right|=O\left(\frac{\log \log P_i}{P_i\log P_i}\right)=o\left(\frac{1}{\sqrt{P_i}\log P_i}\right).$$

These are two coupled integers, possibly equal, linked by the proved divisibility and price comparison. The construction does not assign a sign to either Robin margin, produce a Robin violation, or replace the need to estimate the actual CA candidate. If the starting $C_i$ have $h=1$, their comparators eventually have $h=0$; no infinite such CA subsequence is asserted. Thus a small-benefit relaxation of an $h=1$ optimizing condition requires its own phase transfer instead of assuming that the unit window is preserved. Neither this diagnostic nor layer removal supplies the missing same-source signed prime-error estimate.

The controlled comparator does not enter the existing slow-norm family

The comparator’s eventual unit bit zero does not place it in the known uniformly safe FIB family. This can be checked using existing suppliers, without assuming a Robin violation or introducing a new prime estimate. Write

$$\Delta (n)=e^{\gamma }\log \log n-Z(n).$$

First reuse the same-cutoff envelope calculation in the Nicolas note. For $X\to \infty$, put $v=\log X$ and $L=\log v$. The already inspected Dusart inputs give

$$\log \frac{\Phi (X)}{e^{\gamma }L}=O(L^{-2}),$$

while Nicolas’s Theorem 1.2 at the fixed order $J=1$ gives

$$\log \frac{\Phi (X)}{\Sigma (X)}=O\left(\frac{1}{\sqrt{v}L}\right).$$

Consequently

$$e^{\gamma }\log \log X-\Sigma (X)=O(1/L)⟶0.$$

This is an application of the existing unconditional envelope and prime-product estimates, rather than a new additive asymptotic theorem. The classical record property gives $\Sigma (C)=Z(C)$ for every CA $C$, including tied maximizers. Hence $\Delta (C_i)\to 0$ for every $C_i\to \infty$ used above. A relative Grönwall limit alone would not justify this additive conclusion.

For their actual comparators $N_i=t_i{C}_i$, retain the price bound and the same actual pair, with possible equality. Since $Z(C_i)=O(\log P_i)$, the response change and budget change both satisfy

$$\begin{array}{rl}0\le Z(N_i)-Z(C_i)& =O\left(\frac{\log \log P_i}{P_i}\right),\\ 0\le e^{\gamma }(\log \log N_i-\log \log C_i)& =O\left(\frac{\log \log P_i}{P_i}\right).\end{array}$$

Thus $|\Delta (N_i)-\Delta (C_i)|\to 0$ and $\Delta (N_i)\to 0$ as well. Neither additive limit determines a Robin sign.

Now use the eventual canonical $h(N_i)=0$ decomposition, with the actual nonnegative nonzero composition $(A_i,B_i)$:

$$g_i=\gcd (A_i,B_i),x_i=(A_i/g_i,B_i/g_i),U_i=2(x_i{)}_1+3(x_i{)}_2,D_i=|Q(x_i)|,N_i=g_i{U}_i.$$

The existing canonical-strip bound in FIB theory, Corollary 200.6 gives $g_i<U_i$, so $\sqrt{N_i}<U_i\le N_i$ and $U_i\to \infty$. For any fixed $0<\omega <1/2$, that volume’s §201.6 supplies a uniform positive additive margin on the family

$$\log \log (2+D_i)\le (\log \log \log U_i{)}^{\omega },g_i\le U_i.$$

If these comparators belonged to that family infinitely often, the same actual subsequence would have $\Delta (N_i)\ge \frac{1}{2}{e}^{\gamma }(\log 2-H(\omega ))>0$ eventually, contradicting $\Delta (N_i)\to 0$. Here $H(\omega )=-\omega \log \omega -(1-\omega )\log (1-\omega )$ is binary entropy. Therefore, under the analytic inputs of the existing §201.6 paper derivation, for each fixed $\omega$ the actual comparators eventually obey

$$\log \log (2+D_i)>(\log \log \log U_i{)}^{\omega }.$$

Section 263 already translates failure of this condition into reduced-core and absolute-depth restrictions for hypothetical violations; those calculations need not be repeated. Here the additional application is to the constructed near-CA comparators, without any violation hypothesis. It also applies directly to any subsequence of CA integers having canonical unit bit zero. The thresholds may depend on $\omega$; no uniform conclusion with $\omega ={\omega }_i$ is asserted.

This excludes a proposed route that would force these small-benefit comparators into the already safe slow-norm region. It leaves unrestricted changing norms and the actual signed margin unresolved. The applications above are paper-level combinations of cited results, with no new Lean verification or literature-priority claim.