Explicit small sets with ε-discrepancy on Bohr sets

Adi Akavia

doi:https://doi.org/10.1016/j.ipl.2014.05.002

Explicit small sets with ε-discrepancy on Bohr sets

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Abstract

We prove that there are explicit sets A that have ε-discrepancy on all rank d Bohr sets in ^ZN and are of size |A|=poly(( ^lnN)d,1/ε). This extends the result on explicit sets with ε-discrepancy on arithmetic progressions Razborov et al. [2] (1993) to Bohr sets which are the higher rank analogue of arithmetic progressions. Our proof is via a reduction to the existence of explicit small biased sets in ^ZN.

Original language	American English
Pages (from-to)	564-567
Number of pages	4
Journal	Information Processing Letters
Volume	114
Issue number	10
DOIs	https://doi.org/10.1016/j.ipl.2014.05.002
State	Published - Oct 2014
Externally published	Yes

Keywords

Arithmetic progressions
Bohr sets
Explicit construction
ε-Biased sets
ε-Discrepancy sets

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Signal Processing
Information Systems
Computer Science Applications

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https://doi.org/10.1016/j.ipl.2014.05.002

Cite this

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abstract = "We prove that there are explicit sets A that have ε-discrepancy on all rank d Bohr sets in ZN and are of size |A|=poly(( lnN)d,1/ε). This extends the result on explicit sets with ε-discrepancy on arithmetic progressions Razborov et al. [2] (1993) to Bohr sets which are the higher rank analogue of arithmetic progressions. Our proof is via a reduction to the existence of explicit small biased sets in ZN.",

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AB - We prove that there are explicit sets A that have ε-discrepancy on all rank d Bohr sets in ZN and are of size |A|=poly(( lnN)d,1/ε). This extends the result on explicit sets with ε-discrepancy on arithmetic progressions Razborov et al. [2] (1993) to Bohr sets which are the higher rank analogue of arithmetic progressions. Our proof is via a reduction to the existence of explicit small biased sets in ZN.

KW - Arithmetic progressions

KW - Bohr sets

KW - Explicit construction

KW - ε-Biased sets

KW - ε-Discrepancy sets

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EP - 567

JO - Information Processing Letters

JF - Information Processing Letters

IS - 10

ER -

Explicit small sets with ε-discrepancy on Bohr sets

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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