Addictive Number Theory

Nathanson, Melvyn B.

doi:10.1007/978-0-387-68361-4_1

Cited by 1 publication

(2 citation statements)

References 44 publications

(7 reference statements)

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“…But for every

k ⩾ 1

we have

c q_{k - 1}^{- κ} ⩽ false⟨ q_{k - 1} η false⟩ < q_{k}^{- 1}

, that is,

q_{k} < c^{- 1} q_{k - 1}^{κ}

; hence we get

\begin{matrix} true \\ right \sum_{1 ⩽ j ⩽ q_{ℓ} / 2} \frac{1}{j false⟨ j η false⟩} ≪ c^{- 1} (\log q_{ℓ}) \sum_{k = 1}^{ℓ} q_{k - 1}^{κ - 1} ≪ c^{- 1} {false(\log q_{ℓ} false)}^{2} q_{ℓ - 1}^{κ - 1}, \end{matrix}

where we used the fact that

q_{ℓ}

is bounded below by the

ℓ

th Fibonacci number. Next note that for any

ℓ ⩾ 1

and

h ⩾ 1

, by [28, Lemma 4.9],

\begin{matrix} true \\ right \sum_{h q_{ℓ} + 1 ⩽ j ⩽ (h + 1) q_{ℓ}} \frac{1}{j^{2} + T j ⟨ j η ⟩} ⩽ \frac{1}{T h q_{ℓ}} \sum_{r = 1}^{q_{ℓ}} \min (() \frac{T}{h q_{ℓ}}, \frac{1}{⟨ false(h q_{ℓ} + r false) η ⟩}) ≪ \frac{1}{{(h q_{ℓ})}^{2}} + \frac{\log q_{ℓ}}{T h} \end{matrix}

…”

Section: Linear Form Diophantine Conditionsmentioning

confidence: 99%

“…Let

p_{k} / q_{k}

be the

k

th convergent of the (simple) continued fraction expansion of

η

(see, for example, [14, Chapter X]; in particular

1 = q_{0} ⩽ q_{1} < q_{2} < \dots

). For any

ℓ ⩾ 1

we have

\begin{matrix} true \\ right \sum_{1 ⩽ j ⩽ q_{ℓ} / 2} \frac{1}{j false⟨ j η false⟩} = \sum_{k = 1}^{ℓ} \sum_{q_{k - 1} / 2 < j ⩽ q_{k} / 2} \frac{1}{j false⟨ j η false⟩} ≪ \sum_{k = 1}^{ℓ} q_{k - 1}^{- 1} \sum_{1 ⩽ j ⩽ q_{k} / 2} \frac{1}{false⟨ j η false⟩} ≪ \sum_{k = 1}^{ℓ} \frac{q_{k} \log q_{k}}{q_{k - 1}}, \end{matrix}

where the last bound follows from [28, Lemma 4.8], since

false| η - \frac{p_{k}}{q_{k}} false| < \frac{1}{q_{k} q_{k + 1}}

[14, Theorem 171]. But for every

k ⩾ 1

we have

c q_{k - 1}^{- κ} ⩽ false⟨ q_{k - 1} η false⟩ < q_{k}^{- 1}

, that is,

q_{k} < c^{- 1} q_{k - 1}^{κ}

; hence we get

\begin{matrix} tru... \end{matrix}

…”

Section: Linear Form Diophantine Conditionsmentioning

confidence: 99%

See 1 more Smart Citation

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

Strömbergsson

Vishe

2020

J. London Math. Soc.

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Let G = SL(2, R) ⋉ (R 2 ) ⊕k and let Γ be a congruence subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕k . We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ\G which project to pieces of closed horocycles in SL(2, Z)\ SL(2, R).As an application, we prove an effective quantitative Oppenheim type result for the quadraticfollowing the approach by Marklof [24] using theta sums.1 Apply [5, Thm. 3] with d = 2 and M = 12 and use the anti-automorphism (M, ( x x ′ )) → ( t M, t M x ′ −x ) of G to translate from the setting with G/Γ in [5] into our setting with X = Γ\G. As noted in [5, Remark 7.2], the proof of [5, Thm. 3] extends trivially to the case when Γ is an arbitrary subgroup of SL(2, Z) ⋉ (Z 2 ) ⊕k of finite index.

show abstract

“…But for every

k ⩾ 1

we have

c q_{k - 1}^{- κ} ⩽ false⟨ q_{k - 1} η false⟩ < q_{k}^{- 1}

, that is,

q_{k} < c^{- 1} q_{k - 1}^{κ}

; hence we get

\begin{matrix} true \\ right \sum_{1 ⩽ j ⩽ q_{ℓ} / 2} \frac{1}{j false⟨ j η false⟩} ≪ c^{- 1} (\log q_{ℓ}) \sum_{k = 1}^{ℓ} q_{k - 1}^{κ - 1} ≪ c^{- 1} {false(\log q_{ℓ} false)}^{2} q_{ℓ - 1}^{κ - 1}, \end{matrix}

where we used the fact that

q_{ℓ}

is bounded below by the

ℓ

th Fibonacci number. Next note that for any

ℓ ⩾ 1

and

h ⩾ 1

, by [28, Lemma 4.9],

\begin{matrix} true \\ right \sum_{h q_{ℓ} + 1 ⩽ j ⩽ (h + 1) q_{ℓ}} \frac{1}{j^{2} + T j ⟨ j η ⟩} ⩽ \frac{1}{T h q_{ℓ}} \sum_{r = 1}^{q_{ℓ}} \min (() \frac{T}{h q_{ℓ}}, \frac{1}{⟨ false(h q_{ℓ} + r false) η ⟩}) ≪ \frac{1}{{(h q_{ℓ})}^{2}} + \frac{\log q_{ℓ}}{T h} \end{matrix}

…”

Section: Linear Form Diophantine Conditionsmentioning

confidence: 99%

“…Let

p_{k} / q_{k}

be the

k

th convergent of the (simple) continued fraction expansion of

η

(see, for example, [14, Chapter X]; in particular

1 = q_{0} ⩽ q_{1} < q_{2} < \dots

). For any

ℓ ⩾ 1

we have

\begin{matrix} true \\ right \sum_{1 ⩽ j ⩽ q_{ℓ} / 2} \frac{1}{j false⟨ j η false⟩} = \sum_{k = 1}^{ℓ} \sum_{q_{k - 1} / 2 < j ⩽ q_{k} / 2} \frac{1}{j false⟨ j η false⟩} ≪ \sum_{k = 1}^{ℓ} q_{k - 1}^{- 1} \sum_{1 ⩽ j ⩽ q_{k} / 2} \frac{1}{false⟨ j η false⟩} ≪ \sum_{k = 1}^{ℓ} \frac{q_{k} \log q_{k}}{q_{k - 1}}, \end{matrix}

where the last bound follows from [28, Lemma 4.8], since

false| η - \frac{p_{k}}{q_{k}} false| < \frac{1}{q_{k} q_{k + 1}}

[14, Theorem 171]. But for every

k ⩾ 1

we have

c q_{k - 1}^{- κ} ⩽ false⟨ q_{k - 1} η false⟩ < q_{k}^{- 1}

, that is,

q_{k} < c^{- 1} q_{k - 1}^{κ}

; hence we get

\begin{matrix} tru... \end{matrix}

…”

Section: Linear Form Diophantine Conditionsmentioning

confidence: 99%

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

Strömbergsson

Vishe

2020

J. London Math. Soc.

View full text Add to dashboard Cite

show abstract

Addictive Number Theory

Cited by 1 publication

References 44 publications

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

An effective equidistribution result for SL(2,R)⋉(R2)⊕k and application to inhomogeneous quadratic forms

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