Solution of Cassels' Problem on a Diophantine Constant over Function Fields

Efrat Bank; Erez Nesharim; Steffen Højris Pedersen

doi:10.1093/imrn/rnw169

Solution of Cassels' Problem on a Diophantine Constant over Function Fields

Efrat Bank, Erez Nesharim, Steffen Højris Pedersen

Research output: Contribution to journal › Article › peer-review

Abstract

This article deals with an analogue of Cassels' problem on inhomogeneous Diophantine approximation in function fields. The inhomogeneous approximation constant of a Laurent series θ € F₁((1/ t)) with respect to ? € F₁((1/ t)) is defined to be c(θ, ? ) = inf_0N€Fq[t]= |N| |<Nθ ? γ>|. We show that inf_0N€Fq[t]Fq θ € F_q((1/t )) sup_€((1/t)) c(θ, γ ) = q^-2, and prove that for every θ the set BAθ = {γ € Fq ((1/t)): c(θ, γ ) > 0 } has full Hausdorff dimension. Our methods generalize easily to the case of vectors in F₁ ((1/t))^d.

Original language	English
Pages (from-to)	5451-5474
Number of pages	24
Journal	International Mathematics Research Notices
Volume	2017
Issue number	18
DOIs	https://doi.org/10.1093/imrn/rnw169
State	Published - 2017
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1093/imrn/rnw169

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title = "Solution of Cassels' Problem on a Diophantine Constant over Function Fields",

abstract = "This article deals with an analogue of Cassels' problem on inhomogeneous Diophantine approximation in function fields. The inhomogeneous approximation constant of a Laurent series θ € F1((1/ t)) with respect to ? € F1((1/ t)) is defined to be c(θ, ? ) = inf0N€Fq[t]= |N| ||. We show that inf0N€Fq[t]Fq θ € Fq((1/t )) sup€((1/t)) c(θ, γ ) = q-2, and prove that for every θ the set BAθ = {γ € Fq ((1/t)): c(θ, γ ) > 0 } has full Hausdorff dimension. Our methods generalize easily to the case of vectors in F1 ((1/t))d.",

author = "Efrat Bank and Erez Nesharim and {H{\o}jris Pedersen}, Steffen",

note = "Funding Information: This work was funded by ERC starter grant DLGAPS 279893, BSF grant 2010428 and the Danish Funding Information: Natural Science Research Council.",

year = "2017",

doi = "10.1093/imrn/rnw169",

language = "الإنجليزيّة",

volume = "2017",

pages = "5451--5474",

journal = "International Mathematics Research Notices",

issn = "1073-7928",

publisher = "Oxford University Press",

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T1 - Solution of Cassels' Problem on a Diophantine Constant over Function Fields

AU - Bank, Efrat

AU - Nesharim, Erez

AU - Højris Pedersen, Steffen

N1 - Funding Information: This work was funded by ERC starter grant DLGAPS 279893, BSF grant 2010428 and the Danish Funding Information: Natural Science Research Council.

PY - 2017

Y1 - 2017

N2 - This article deals with an analogue of Cassels' problem on inhomogeneous Diophantine approximation in function fields. The inhomogeneous approximation constant of a Laurent series θ € F1((1/ t)) with respect to ? € F1((1/ t)) is defined to be c(θ, ? ) = inf0N€Fq[t]= |N| ||. We show that inf0N€Fq[t]Fq θ € Fq((1/t )) sup€((1/t)) c(θ, γ ) = q-2, and prove that for every θ the set BAθ = {γ € Fq ((1/t)): c(θ, γ ) > 0 } has full Hausdorff dimension. Our methods generalize easily to the case of vectors in F1 ((1/t))d.

AB - This article deals with an analogue of Cassels' problem on inhomogeneous Diophantine approximation in function fields. The inhomogeneous approximation constant of a Laurent series θ € F1((1/ t)) with respect to ? € F1((1/ t)) is defined to be c(θ, ? ) = inf0N€Fq[t]= |N| ||. We show that inf0N€Fq[t]Fq θ € Fq((1/t )) sup€((1/t)) c(θ, γ ) = q-2, and prove that for every θ the set BAθ = {γ € Fq ((1/t)): c(θ, γ ) > 0 } has full Hausdorff dimension. Our methods generalize easily to the case of vectors in F1 ((1/t))d.

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U2 - 10.1093/imrn/rnw169

DO - 10.1093/imrn/rnw169

M3 - مقالة

SN - 1073-7928

VL - 2017

SP - 5451

EP - 5474

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 18

ER -

Solution of Cassels' Problem on a Diophantine Constant over Function Fields

Abstract

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