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Definition 4.2

Define \(a_i,b_i,c_i\in \mathbb {R}_{\geq 0}\) via

\[ X_i=X^{a_i}, \quad Y_i=X^{b_i}, \quad Z_i=X^{c_i}, \]

for \(1\leq i\leq d\), and \(a_i=b_i=c_i=0\) for \(i{\gt}d\).

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Definition 2.4

For \(\mathbf{c}\in \mathbb {Z}^3\) and \(\mathbf{X},\mathbf{Y},\mathbf{Z}\in \mathbb {R}_{{\gt}0}^{d}\). we have

\begin{equation} \label{eq:Bk} B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}) := \# \left\{ (\mathbf{x},\mathbf{y},\mathbf{z})\in \mathbb {N}^{3d}\; :\; \begin{array}{l} x_i\sim X_i,\, y_i\sim Y_i,\, z_i\sim Z_i \\ c_1\prod _{j\leq d}x_j^j+c_2\prod _{j\leq d}y_j^j=c_3\prod _{j\leq d}z_j^j\\ \gcd (c_1\prod _{j\leq d}x_j,c_2\prod _{j\leq d}y_j , c_3\prod _{j\leq d}z_j)=1 \end{array} \right\} . \end{equation}

2.0.3

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Lean

B
B_finset

Definition 4.11

It will be convenient to define constants \(\delta _a,\delta _b,\delta _c\) via

\begin{equation} \label{eq:ai1/3} \sum _{i\leq d} a_i \, = \frac{1}{3}-\delta _a, \quad \sum _{i\leq d} b_i \, = \frac{1}{3}-\delta _b, \quad \sum _{i\leq d} c_i = \frac{1}{3}-\delta _c, \end{equation}

4.3.1

together with

\begin{equation*} \delta _{ab}:=\delta _a+\delta _b,\quad \delta _{ac}:=\delta _a+\delta _c, \quad \delta _{bc}:=\delta _b+\delta _c, \end{equation*}

and \(\delta _s := \delta _a+\delta _b+\delta _c\).

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Lean

δ_
delta_s

Definition 1.1

For \(\lambda {\gt}0\) define \(N_\lambda (X)\) as the number of triples \((a,b,c)\in \mathbb {N}^3\) with \(a+b=c\), \(\gcd (a,b,c)=1\) and \(\operatorname{rad}(abc) {\lt} c^{\lambda }\).

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Definition 4.5

It will be convenient to henceforth define

\begin{equation*} \nu =2\varepsilon ^2+\frac{\log B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z})}{\log X}. \end{equation*}

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Definition 3.10

Let \(\omega (n)\) denote the number of distinct prime factors of an integer \(n\).

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Definition 4.13

Define \(s_i := a_i + b_i + c_i\).

LaTeX Lean

Definition 1.5

For \(\alpha ,\beta ,\gamma {\gt}0\), define \(S_{\alpha ,\beta ,\gamma }(X)\) as the number of \((a,b,c)\in \mathbb {N}^3\) with \(\gcd (a,b,c)=1\) such that

\[ a,b,c\in [1,X], \quad a+b=c, \quad \operatorname{rad}(a)\leq X^\alpha , \quad \operatorname{rad}(b)\leq X^\beta , \quad \operatorname{rad}(c)\leq X^\gamma . \]

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Definition 2.1

Let \(S^*_{\alpha ,\beta ,\gamma }(X)\) to be the number of \((a,b,c)\in \mathbb {N}^3\) with \(\gcd (a,b,c)=1\) and

\begin{align*} c\in [X/2,X],\quad a+b=c,\quad \operatorname{rad}(a)\sim X^{\alpha },\quad \operatorname{rad}(b)\sim X^{\beta },\quad \operatorname{rad}(c)\sim X^{\gamma }. \end{align*}

LaTeX Lean

Lemma 4.3

\begin{equation} \label{eq:oat1} \sum _{i\leq d} ia_i \leq 1, \quad \sum _{i\leq d} ib_i \leq 1, \quad 1-\varepsilon ^2\leq \sum _{i\leq d} ic_i \leq 1. \end{equation}

4.1.2

LaTeX Lean

Lemma 4.21

\(\nu \leq 0.66\) in the case \(s_2 \geq 0.3\).

LaTeX Lean

Lemma 4.18

If \(s_2 \geq 0.3\),

\begin{equation} \label{eq:s1} s_1 \leq 0.04+\delta \end{equation}

4.3.17

and \(s_4 {\lt} 0.21 +\frac{3}{2}\delta .\)

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Lean

bound_4_point_22
case_1_helper

Lemma 4.24

Assuming 4.27, 4.28, 4.29, 4.30, \(\nu \leq 0.66\) in the case \(s_2 {\lt} 0.3\).

LaTeX Lean

Lemma 4.22

\begin{equation} \label{eq:2b3} b_3 {\lt} 0.17+\frac{\delta }{2}. \end{equation}

4.3.21

LaTeX Lean

Lemma 4.23

\begin{equation} \label{eq:Case2Basic2} a_3 \geq 0.32 - 4 \delta _s - s_1 - 2 \delta . \end{equation}

4.3.23

LaTeX Lean

Lemma 1.2

For any \(\varepsilon {\gt}0\), we have

\begin{equation} \label{eq:bruijn} \# \left\{ n\leq x : \operatorname{rad}(n)\leq x^\lambda \right\} \ll _\varepsilon x^{\lambda +\varepsilon }. \end{equation}

1.0.1

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Lemma 4.12

We have

\begin{equation} \label{eq:robin2} \delta _{ab}, \delta _{ac}, \delta _{bc} \leq 0.00\overline{6} + \varepsilon ^2, \end{equation}

4.3.2

\begin{equation} \label{eq:robin1} -0.00\bar6 - \delta \leq \delta _a,\delta _b,\delta _c \leq 0.01\bar3+\delta +\varepsilon , \end{equation}

4.3.3

and

\begin{equation} \label{eq:robin3} -\delta {\lt} \delta _s \leq 0.01+\varepsilon . \end{equation}

4.3.4

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bound_4_point_8
bound_4_point_9_lower
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bound_4_point_10_lower
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Lemma 2.5

Let \(\varepsilon \in (0,1/2)\), and let \(2\leq n\leq X\) be an integer. Then there exists a factorisation

\begin{align*} n=c\prod _{j\leq \frac{5}{2}\varepsilon ^{-2}}x_j^{j}, \end{align*}

for positive integers \(x_j,c\) such that \(c\leq X^{\varepsilon /2}\), the \(x_j\) are pairwise coprime, and

\begin{align*} X^{-\varepsilon }\prod _{j\leq \frac{5}{2}\varepsilon ^{-2}}x_j \leq \operatorname{rad}(n) \leq X^{\varepsilon }\prod _{j\leq \frac{5}{2}\varepsilon ^{-2}}x_j. \end{align*}

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Lean

exists_nice_factorization
exists_nice_factorization'

Lemma 3.8

Let \(\varepsilon {\gt}0\). Let \(D\ge 1\) be a squarefree integer, and set \(K=\mathbb {Q}(\sqrt{-D})\). Then for all \(\alpha \in K\), the number of ideals dividing \((\alpha )\) in \(K\) is \(O(N_\alpha ^\varepsilon )\).

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Lemma 3.4

Let \(r\geq 1\) and let \(g\in \mathbb {C}[x]\) be a polynomial which has at least one root of multiplicity \(1\). Then the polynomial \(g(x)-y^r\) is absolutely irreducible.

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Lemma 3.12

Let \(\varepsilon {\gt}0\) and \(D\geq 1\) and assume that \(p,q,r\in [1,D]\) are integers. Then

\[ N(X,Y,Z)\ll _{\varepsilon ,D} Z \min \left(X^{\frac{1}{q}}, Y^{\frac{1}{p}} \right) (XY)^\varepsilon , \]

where the implied constant only depends on \(\varepsilon \) and \(D\). Furthermore, if \(p=q\geq 2\), then we have

\[ N(X,Y,Z)\ll _{\varepsilon ,D} Z (|a_1a_2a_3|XYZ)^\varepsilon . \]

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Lemma 3.13

Let \(\varepsilon {\gt}0\) and \(D\geq 1\) and assume that \(p=q,r\in [1,D]\) are integers. Then

\[ N(X,Y,Z)\ll _{\varepsilon ,D} Z (|a_1a_2a_3|XYZ)^\varepsilon . \]

where the implied constant only depends on \(\varepsilon \) and \(D\).

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Lemma 2.2

We have

\begin{equation} \label{eq:step1} N_\lambda (X)\ll (\log X)^4 \max _{\substack { \alpha ,\beta ,\gamma {\gt}0\\ \alpha +\beta +\gamma \leq \lambda }} S^*_{\alpha ,\beta ,\gamma }(X). \end{equation}

2.0.1

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Lean

countTriples_isBigO_dyadicSup
countTriples_le_log_pow_mul_sup

Lemma 3.11

For any \(n\ge 2\), we have \(\omega (n) \ll \log (3n)/\log \log (3n)\).

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Lemma 4.15

We may assume

\begin{equation} \label{eq:SpecialGeometry} s_1+s_2\leq 0.34 +\delta . \end{equation}

4.3.10

LaTeX Lean

Lemma 4.19

\(\nu \leq 0.66\) in the case \(s_2 \geq 0.3\) and \(b_3\le 0.34-s_1-s_2+\delta \).

LaTeX Lean

Lemma 4.20

\(\nu \leq 0.66\) in the case \(s_2 \geq 0.3\) and \(b_3 {\gt} 0.34-s_1-s_2+\delta \).

LaTeX Lean

Lemma 4.25

\(\nu \leq 0.66\) in the case \(s_2{\lt}0.3\) and \(a_3\geq 0.32\).

LaTeX Lean

Lemma 4.26

\(\nu \leq 0.66\) in the case \(s.2 {\lt} 0.3\) and \(b_3+c_3{\lt}0.33-\frac{s_2}{2}-\frac{\delta }{2}\).

LaTeX Lean

Lemma 4.27

\(\nu \leq 0.66\) in the case \(s.2 {\lt} 0.3\) and \(4s_1+3s_2{\gt}0.71\).

LaTeX Lean

Lemma 4.28

\(\nu \leq 0.66\) in the case \(s.2 {\lt} 0.3\) and \(4s_1+s_2{\lt}0.4\).

LaTeX Lean

Lemma 4.29

\(\nu \leq 0.66\) in the case \(s.2 {\lt} 0.3\) and \(0.066\leq s_2\leq 0.204\).

LaTeX Lean

Lemma 4.30

\(\nu \leq 0.66\) in the case \(s_2{\lt}0.3\) and \(2s_1-s_2{\gt}0.025\).

LaTeX Lean

Lemma 4.16

For any \(j\ge 3\), allow \(\tau _j\) to be an element

\begin{equation} \label{eq:taujdef} \tau _j\in \{ a_j,b_j,c_j, s_j, a_j+b_j,a_j+c_j,b_j+c_j\} . \end{equation}

4.3.11

Then

\begin{equation} \label{eq:Geotauj} \tau _j\in \left(0.34-s_1-s_2+\delta , \frac{0.66 -s_2-\delta }{j-1}\right) \Longrightarrow \nu {\lt} 0.66 \end{equation}

4.3.12

and

\begin{equation} \label{eq:Geotau3} \tau _3\in \left(0.34-s_1+\delta , 0.33-\frac{1}{2}\delta \right) \Longrightarrow \nu {\lt} 0.66. \end{equation}

4.3.13

LaTeX Lean

Lemma 4.17

We have

\begin{equation} \label{eq:black1} a_3 \ge \frac{1}{3}-4\delta _a - 3a_1 - 2a_2, \end{equation}

4.3.14

\begin{equation*} a_3 \ge \frac{1}{3}-\frac{5}{2}\delta _a - 2a_1 - \frac{3}{2}a_2 - \frac{1}{2}a_4. \end{equation*}

Therefore

\begin{equation} s_3 \ge 1-4\delta _s - 3s_1 - 2s_2, \label{eq:s3{\gt}s12} \end{equation}

4.3.15

and

\begin{equation} s_3 \ge 1-\frac{5}{2}\delta _s - 2s_1 - \frac{3}{2}s_2 - \frac{1}{2}s_4. \label{eq:s3{\gt}s124} \end{equation}

4.3.16

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bound_4_point_19_first
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Lemma 4.4

We have

\begin{equation} \label{eq:oat4} \sum _{i\leq d} (a_i+b_i)\geq 0.66-\varepsilon ^2,\quad \sum _{i\leq d} (a_i+c_i)\geq 0.66-\varepsilon ^2,\quad \sum _{i\leq d} (b_i+c_i) \geq 0.66-\varepsilon ^2 \end{equation}

4.1.3

and

\begin{equation} \label{eq:oat3} \sum _{i\leq d} (a_i+b_i+c_i) \leq 1+\delta -\varepsilon . \end{equation}

4.1.4

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Lean

Bound4Point3
Bound4Point4

Lemma 1.6

\begin{equation} \label{eq:trivial} S_{\alpha ,\beta ,\gamma }(X)\ll _\varepsilon X^{\min \{ \alpha +\beta , \alpha +\gamma , \beta +\gamma \} +\varepsilon }, \end{equation}

1.0.3

for any \(\varepsilon {\gt}0\).

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Proposition 1.3

Let \(\lambda {\gt}0\). Then \(N_\lambda (X)=O_\varepsilon ( X^{2\lambda /3+\varepsilon })\), for any \(\varepsilon {\gt}0\).

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Proposition 4.9Determinant Bound

\begin{align*} \nu {\lt} \min _{p,q\ge 1} \left( 1+\delta - a_p - b_q +\min \left(\frac{a_p}{q}, \frac{b_q}{p}\right)\right). \end{align*}

LaTeX Lean

Proposition 3.14

Let \(d\geq 1\), and let

\[ X_1,\ldots , X_d, Y_1,\ldots , Y_d,Z_1,\ldots , Z_d\geq 1. \]

Let \(\mathbf{c}\in (c_1,c_2,c_3)\in \mathbb {Z}_{\neq 0}^3\). Then for \(\Delta \) as in 3.1.1, we have

\[ B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}) \ll \Delta ^\varepsilon \prod _{j\leq d} X_j Y_j Z_j \cdot \min _{p,q\geq 1}\left( (X_p Y_q)^{-1} \min \left( X_p^{\frac{1}{q}} , Y_q^{\frac{1}{p}}\right)\right). \]

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Proposition 2.6

Let \(\alpha ,\beta ,\gamma \in (0,1]\) be fixed and let \(X\geq 2\). For any \(\varepsilon {\gt}0\) there exists an integer \(d=d(\varepsilon )\geq 1\) such that the following holds. There exist \(X_1,\ldots , X_d,Y_1,\ldots , Y_d, Z_1,\ldots , Z_d\geq 1\) satisfying

\begin{align} \label{eq:xiyizi_1} X^{\alpha -\varepsilon }\ll _{\varepsilon } \prod _{j=1}^{d}X_j\leq 2 X^{\alpha +\varepsilon },\quad X^{\beta -\varepsilon }\ll _{\varepsilon }\prod _{j=1}^{d}Y_j\leq 2 X^{\beta +\varepsilon },\quad X^{\gamma -\varepsilon }\ll _{\varepsilon }\prod _{j=1}^{d}Z_j\leq 2 X^{\gamma +\varepsilon } \end{align}

and

\begin{align} \label{eq:xiyizi_2} \prod _{j=1}^d X_j^j \leq X, \quad \prod _{j=1}^d Y_j^j\leq X,\quad X^{1-\varepsilon ^2}\ll _{\varepsilon } \prod _{j=1}^d Z_j^j\leq X \end{align}

and pairwise coprime integers \(1\leq c_1,c_2,c_3\leq X^{\varepsilon }\), such that

\begin{align*} S^*_{\alpha ,\beta ,\gamma }(X) \ll _{\varepsilon } X^{\varepsilon }B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}). \end{align*}

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Proposition 4.6

\begin{equation} \label{eq:oat5} 0.32 - \delta \leq \sum _{i\leq d} a_i ,~ \sum _{i\leq d} b_i ,~ \sum _{i\leq d} c_i \leq 0.34 + \delta - \frac{1}{2}\varepsilon , \end{equation}

4.1.5

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Proposition 3.1Fourier analysis bound

Let \(d\geq 1\), \(\varepsilon {\gt}0\) and \(A\geq 1\) be fixed. Let

\[ X_1,\ldots , X_d, Y_1,\ldots , Y_d,Z_1,\ldots , Z_d\geq 1 \]

and put

\begin{equation} \label{eq:Delta} \Delta =\max _{1\leq i\leq d}(X_i Y_i Z_i). \end{equation}

3.1.1

Let \(\mathbf{c}=(c_1,c_2,c_3)\in \mathbb {Z}^3\) satisfy \(0{\lt}|c_1|,|c_2|,|c_3|\leq \Delta ^A\). Then

\begin{align*} B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}) \ll \Delta ^\varepsilon \frac{\prod _{j\leq d} \big(X_j Y_j Z_j(Y_j+Z_j)\big)^{\frac{1}{2}}}{\max _{i{\gt}1}\prod _{j\equiv 0\bmod i}Z_j^{\frac{1}{2}}}. \end{align*}

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Proposition 4.7Fourier bound

\begin{align*} \nu & {\lt} \frac{1}{2}\Big(1+\delta +\sum _{i\leq d} \max (a_i,b_i) - \max _{m{\gt}1}(a_m,b_m)\Big). \end{align*}

LaTeX Lean

Proposition 4.8Geometry bound

\begin{align*} \nu {\lt} \delta + \min _{I,I',I''\subset [d]} \left( \max \left( 1 \, ,\, \sum _{i\in I} ia_i +\sum _{i\in I'} ib_i + \sum _{i\in I''} ic_i\right) - \sum _{i\in I} a_i -\sum _{i\in I'} b_i - \sum _{i\in I''} c_i\right). \end{align*}

LaTeX Lean

Proposition 3.2Geometry of numbers bound

Let \(d\geq 1\) and \(\varepsilon {\gt}0\) be fixed, and let

\[ X_1,\ldots , X_d, Y_1,\ldots , Y_d,Z_1,\ldots , Z_d\geq 1. \]

Let \(\mathbf{c}\in (c_1,c_2,c_3)\in \mathbb {Z}^3\) have non-zero and pairwise coprime coordinates. Then for \(\Delta \) as in 3.1.1,

\[ B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}) \ll \hspace{-0.1cm} \Delta ^{\varepsilon } \hspace{-0.3cm} \min _{I,I',I''\subset [d]} \hspace{-0.1cm} \Big(\prod _{i\in I}X_i\prod _{i\in I'}Y_i\prod _{i\in I''}Z_i\Big)\bigg(1 + \frac{\prod _{i\notin I}X_i^i\prod _{i\notin I'}Y_i^i\prod _{i\notin I''}Z_i^i}{\max \{ |c_1|\prod _{i}X_i^i,\; |c_2|\prod _{i}Y_i^i,\; |c_3|\prod _{i}Z_i^i\} }\bigg). \]

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Proposition 4.14

To show \(\nu \leq 0.66\), it suffices to assume

\begin{equation} \label{eq:Thue} a_j+b_j, \, a_j+c_j,\, b_j+c_j {\lt} 0.34+\delta , \end{equation}

4.3.6

for each \(j\geq 2\), and moreover,

\begin{equation} \label{eq:Thue2} a_2+a_4+b_2+b_4, \ a_2+a_4+c_2+c_4, b_2+b_4+c_2+c_4 {\lt} 0.34+\delta . \end{equation}

4.3.7

Hence

\begin{equation} \label{eq:sThue} s_5,s_3,s_2+s_4 {\lt} 0.51+\frac{3}{2}\delta . \end{equation}

4.3.8

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bound_4_point_12
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Proposition 4.10Thue bound

\begin{align*} \nu {\lt} 1 +\delta - \max _{p\ge 2}\sum _{p\mid i}(a_i+b_i). \end{align*}

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Proposition 3.15

Let \(d\geq 1\), and let

\[ X_1,\ldots , X_d, Y_1,\ldots , Y_d,Z_1,\ldots , Z_d\geq 1. \]

Let \(\mathbf{c}\in (c_1,c_2,c_3)\in \mathbb {Z}_{\neq 0}^3\). Then

\[ B_d(\mathbf{c},\mathbf{X},\mathbf{Y},\mathbf{Z}) \ll \Delta ^\varepsilon \prod _{j\leq d} X_j Y_j Z_j \cdot \min _{p\geq 2} \prod _{\substack {j\leq d \\ p\mid j}} (X_j Y_j)^{-1}, \]

where \(\Delta \) is given by 3.1.1.

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Proposition 4.1

Let \(\alpha ,\beta ,\gamma {\gt}0\), and let \(\varepsilon {\gt}0\) be fixed. Then

\[ S^*_{\alpha ,\beta ,\gamma }(X)\ll X^{0.66-\varepsilon ^2/2}, \]

unless \(\min \{ \alpha +\beta , \beta +\gamma ,\gamma +\alpha \} \geq 0.66-\varepsilon ^2\).

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Theorem 1.4

Let \(\lambda \in (0, 1.001)\) be fixed. Then \(N_\lambda (X)=O( X^{33/50})\).

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Theorem 4.31

\(\nu \le 0.66\).

LaTeX Lean

Theorem 2.3

There exists \(\varepsilon {\gt}0\) such that for all \(\mathbf{c}\in \mathbb {Z}^3\) and \(\mathbf{X},\mathbf{Y},\mathbf{Z}\in \mathbb {R}_{{\gt}0}^{d}\). we have

\begin{align} B_d({\bf c, X,Y,Z}) \ll X^{0.66-\varepsilon }. \end{align}

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Theorem 3.5

Let \(f(x)\) be a \(C^\infty \) function on a closed subinterval of \([0,N]\), and suppose that\(F (x, f ) = 0\), where \(F(x, y) \in \mathbb {R}[x, y]\) is absolutely irreducible of degree \(d \ge 2\). Suppose that \(|f'(x)| \le 1\). Then

\begin{align*} \# \{ (x,f(x))\in \{ 1,\ldots ,N\} ^2 \} \ll _d (\log N)^{O(N)} N^{1/d} \end{align*}

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Theorem 3.9

Given integers \(n,a_1,a_2\neq 0\) and \(p\ge 3\), there are at most \(O(p^{1+\omega (|n|)})\) many solutions \((x_1,x_2)\in \mathbb {Z}^2\) such that \(|x_i|\le X_i\) and

\begin{align*} n = a_1 x_1^p + a_2 x_2^p. \end{align*}

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Theorem 3.3

Let \(\gcd (a_1,a_2,a_3)=1\). Then for \(X_1,X_2,X_3{\gt}1\), we have

\begin{align*} \# \left((x_1,x_2,x_3)\in \mathbb {Z}^3 \; :\; \gcd (x_1,x_2,x_3)=1,\, |x_i| \le X_i ,\, a_1x_1+a_2x_2+a_3x_3=0\right) \ll \ 1 \ + \ & \frac{X_1X_2X_3}{\max _i\{ |a_i|X_i\} }. \end{align*}

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Theorem 3.6

Let \(F\in \mathbb {Z}[x_1 ,\ldots , x_n]\) be an absolutely irreducible polynomial of degree \(d\), and let \(\varepsilon {\gt} 0\) and \(B_1,\ldots ,B_n \ge 1\) be given. Define

\begin{align*} B = \max _{(e_1,\ldots ,e_n)}\Big( \prod _{i\le n} B_i^{e_i} \Big) \end{align*}

where the maximum is taken over all integer \(n\)-tuples \((e_1,\ldots ,e_n)\) for which the corresponding monomial \(x_1^{e_1}\cdots x_n^{e_n}\) occurs in \(F\) with non-zero coefficient.

Then there exists \(D = D(n,d,\varepsilon )\) and an integer \(k\) with

\begin{align*} k \ll _{n,d,\varepsilon } T^\varepsilon (\log \| F\| )^{2n-3} \exp \Big((n-1)\Big(\frac{\prod _{i\le n}\log B_i}{\log B}\Big)^{1/(n-1)}\Big) \end{align*}

satisfying the following: There are \(k\) polynomials \(F_1,\ldots ,F_k\in \mathbb {Z}[x_1 ,\ldots , x_n]\) coprime to \(F\), with \(\deg F_i \le D\), such that every root of \(F(x_1 ,\ldots , x_n)=0\) with \(x_i \le B_i\) is also a root of \(F_j(x_1 ,\ldots , x_n)=0\), for some \(j\le k\).

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Theorem 3.7

Given integers \(n,a_1,a_2\neq 0\), there are at most \(O_{\varepsilon ,D}(|na_1a_2 X_1 X_2|^{\varepsilon })\) many solutions \((x_1,x_2)\in \mathbb {Z}^2\) such that \(|x_i|\le X_i\) and

\begin{align*} n = a_1 x_1^2 + a_2 x_2^2. \end{align*}

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