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Reducibility and nonreducibility between ℓ^{ } equivalence relations
We show that, for 1 ≤ p > q > ∞ 1 \le p > q > \infty , the relation of ℓ p \ell ^{p} -equivalence between infinite sequences of real numbers is Borel reducible to the relation of ℓ q \ell ^{q} -equivalence (i.e., the Borel cardinality of the quotient R N / ℓ p {\mathbb {R}}^{\mathbb {N}}...
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| Published in: | Transactions of the American Mathematical Society 1999-05, Vol.351 (5), p.1835-1844 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We show that, for
1
≤
p
>
q
>
∞
1 \le p > q > \infty
, the relation of
ℓ
p
\ell ^{p}
-equivalence between infinite sequences of real numbers is Borel reducible to the relation of
ℓ
q
\ell ^{q}
-equivalence (i.e., the Borel cardinality of the quotient
R
N
/
ℓ
p
{\mathbb {R}}^{\mathbb {N}}/\ell ^{p}
is no larger than that of
R
N
/
ℓ
q
{\mathbb {R}}^{\mathbb {N}}/\ell ^{q}
), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but ‘modular,’ and using this nicer map to derive a contradiction. |
|---|---|
| ISSN: | 0002-9947 1088-6850 |
| DOI: | 10.1090/S0002-9947-99-02261-8 |