bibkey: bretechedrappeau2020smoothaffine authors: Régis de la Bretèche; Sary Drappeau year: 2020 title: Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables doi: 10.4171/JEMS/951 url: https://doi.org/10.4171/JEMS/951 claim: Theorem 4.1 and Corollary 4.2 bound simultaneous smooth affine values with growing coefficients; the actual FIB exception budget pays the cofactor coefficient height, but supplies neither smoothness of n-2 nor a pointwise discriminant or Euler-budget estimate. strata_touched: [] license: citation-only triage: anchor

Simultaneous smooth affine values and the actual norm cofactor

The paper appeared in Journal of the European Mathematical Society 22 (2020), 1577–1624. The inspected primary text is the author-hosted manuscript, whose title page says 25 March 2025. The theorem locators below refer to that version’s printed pages. Its statements and the relevant proof opening were inspected; the full analytic proofs have not been independently verified. The parameter applications below are paper derivations, with no Lean verification or originality claim.

The joint counting theorem and its conditions

Write $\Psi (X,y)=\#\{r\le X:P^{+}(r)\le y\}$ and let $\rho$ be the Dickman function. Theorem 4.1, printed p.13, states that for each fixed $\epsilon >0$ there exist $C,\kappa >0$ such that

$$\#\{r\in [X,(1+\eta )X]:P^{+}(ar+b)\le y_1,P^{+}(c_{0}r+d)\le y_2\}{\ll }_{\epsilon }\eta \Psi (X,y_1)\rho {\left(\frac{\log X}{\log y_2}\right)}^{3/5-\epsilon },\text{\hspace{1em}(S1)}$$

provided

$$\begin{array}{c}|a|,|b|,|c_0|,|d|\le X^{\kappa },a,c_0>0,(a,c_0)=1,ad-b{c}_0\ne 0,\\ (\log X{)}^C\le y_1\le y_2\le X,y_2\le y_1^C,\\ e^{-\kappa \sqrt{\log y_1}}\le \eta \le 1.\end{array}$$

Corollary 4.2, printed p.14, supplies

$$\#\{r\in [X,(1+\eta )X]:P^{+}(r(ar+b))\le y\}{\ll }_{\epsilon }\eta X\rho {\left(\frac{\log X}{\log y}\right)}^{8/5-\epsilon }\text{\hspace{1em}(S2)}$$

for $(\log X{)}^C\le y\le X$, $e^{-\kappa \sqrt{\log y}}\le \eta \le 1$, and $1\le a,|b|\le X^{\kappa }$. Neither statement includes the canonical FIB lift, a character-conductor weight, or an extremal exponent vector.

Extracting the actual supported square depth

Reuse the unit-bit-one exception construction: $n>2$, $n=1+2A+3B$, $c=4A+7B$, nonsquare signed $D=\delta s^2$, $q=|\delta |$, and

$$c^2-\delta s^2=5n(n-2),T=T_1\mid \gcd (c,s).$$

The actual depth definitions also give $T^2\mid 5n$. For an included prime $p\ne 5$, $2b_p=e_p$; at five, $2b_5=e_5+1$. Set

$$g_5=\gcd (T^2,5)\in \{1,5\},L=\frac{T^2}{g_5},a_0=\frac{5}{g_5},k=\frac{n}{L},v=\frac{c}{T},u=\frac{s}{T}.$$

These are integers. In fact $L$ removes the entire primary component $p^{e_p}$ at each selected exception prime, so $(L,k)=1$. Dividing the same norm identity by $T^2$ gives

$$v^2-\delta u^2=a_{0}k(Lk-2),n=Lk.\text{\hspace{1em}(S3)}$$

The two affine forms $a_{0}k$ and $Lk-2$ have $(a_0,L)=1$, determinant $-2a_0\ne 0$, and $\gcd (a_{0}k,Lk-2)\mid 2a_0\mid 10$. Thus (S1) applies with $(a,b,c_0,d)=(a_0,0,L,-2)$ whenever its remaining conditions hold; (S2) uses $(a,b)=(L,-2)$.

For a fixed $T$ stratum and actual integers $n\in [N_{\ast },(1+\eta )N_{\ast }]$, take $X=N_{\ast }/L$. Under the additional SA support asymptotic $\mathrm{rad}(n)=n^{1-o(1)}$, the existing exception budget gives $L\le T^2\le H_1^2=N_{\ast }^{o(1)}$. Consequently $L\le X^{\kappa }$ eventually for the fixed $\kappa >0$ of (S1). The coefficient-height condition is paid by this exponent budget. For quantitative CA input, reuse the same-price excess bound.

This coordinate change keeps the original integer. Its finite-power deficit is still

$$D_{\mathrm{pow}}(n)=-\sum _{p\mid Lk}\log \left(1-p^{-(v_p(L)+v_p(k)+1)}\right),\text{\hspace{1em}(S4)}$$

and is not replaced by $D_{\mathrm{pow}}(k)$. The actual tests defining ${\mathcal{E}}_1$ and the canonical lift must also be retained. For example, a strip condition $|c-\sqrt{5}(n-1)|\le K$ becomes $|Tv-\sqrt{5}(Lk-1)|\le K$, with width $O(1/T)$ in $v$.

Three conditions remain unpaid. Smoothness of $n$ controls $a_{0}k$ apart from a fixed factor five, but says nothing about $n-2=Lk-2$. Taking $y_2$ of size $n$ violates $y_2\le y_1^C$ at logarithmic $y_1$. Also, $P^{+}(n)\asymp \log n$ does not by itself meet the source’s sufficiently large power $(\log X{)}^C$: enlarging $y_1$ counts a larger population. Finally, a count in each fixed stratum, even with both factors smooth, does not give a pointwise deficit or conductor bound at its actual extremal member.

The residual square depth and weighted exception target

The square depth still missing after this extraction is $u=s/T_1$. Write $R=R_1$ and $g_q=\gcd (q,R)$. The exact modulus identities are

$$|D|=q{T}_1^2{u}^2,m_1=\frac{qR}{g_q}=\frac{|D|R}{g_q{T}_1^2{u}^2}.$$

For fixed admissible character-cutoff exponent $b>0$ and $Y=\log n$,

$$m_1^b\le Y⟺u^2\ge \frac{|D|R}{g_q{T}_1^2{Y}^{1/b}}.\text{\hspace{1em}(S5)}$$

This equivalence supplies no estimate for $u$. If additionally $|D|\ge n^{\zeta }$ with fixed $\zeta >0$ and $T_1=n^{o(1)}$, meeting (S5) requires $u\ge n^{\zeta /2-o(1)}$. No such discriminant lower bound is assumed for the actual remaining candidates. The residual factor can contain extra depth at already supported primes; it need not be supported outside $n$. The character theorem’s onset and supplied weight must still be checked after its cutoff is met.

An alternative keeps the primitive character and pays its actual exceptions in weight. Set $w_p=\log (p/(p-1))$ and

$$W_{\delta }(Y)=\sum _{\begin{array}{c}p\le Y\\ {\chi }_{\delta }(p)=-1\end{array}}{w}_p,E_{\delta }(n;Y)=\sum _{\begin{array}{c}p\le Y\\ p\mid n\\ {\chi }_{\delta }(p)=-1\end{array}}{w}_p.$$

Then $W_{\delta }(Y)-E_{\delta }(n;Y)\le J_{\mathrm{miss}}(n;Y)$. The sufficient joint target remains

$$W_{\delta }(Y)-E_{\delta }(n;Y)+D_{\mathrm{pow}}(n)>M(Y)+J_{\mathrm{large}}(n;Y).\text{\hspace{1em}(S6)}$$

These are the same actual Euler-budget terms as in the character-budget note; the signed Mertens term has not been replaced by an absolute value. The actual negative-support set in $E_{\delta }$ can be smaller than the computable superset ${\mathcal{E}}_1$. Charging all of that superset is conservative and can lose positive or zero-character members.

Permitted exponent depth alone cannot pay this subtraction by a uniform local multiple of the finite-power deficit. For $d_p(e)=-\log (1-p^{-e-1})$, elementary logarithm bounds give

$$\frac{w_p}{d_p(e)}\ge p^e-\frac{1}{p}.\text{\hspace{1em}(S7)}$$

At a possible negative support prime other than five, $e=2b_p$; at five, $e=2b_5-1$. Thus a successful use of (S6) still needs a joint restriction on which exceptions occur and on the rest of the actual exponent vector. The joint smooth-affine counting theorem supplies neither (S5) nor (S6) for every actual candidate. Unit bit zero and square $D$ remain separate branches of the general Robin problem.