11 research outputs found
Locally decodable codes and the failure of cotype for projective tensor products
It is shown that for every there exists a Banach space
of finite cotype such that the projective tensor product \ell_p\tp X fails to
have finite cotype. More generally, if satisfy
then
\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a
proved via a connection to the theory of locally decodable codes
High-entropy dual functions over finite fields and locally decodable codes
We show that for infinitely many primes p, there exist dual functions of order k over Fnp that cannot be approximated in L∞-distance by polynomial phase functions of degree k−1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L∞-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences
Gaussian width bounds with applications to arithmetic progressions in random settings
Motivated by problems on random differences in Szemer\'{e}di's theorem and on
large deviations for arithmetic progressions in random sets, we prove upper
bounds on the Gaussian width of point sets that are formed by the image of the
-dimensional Boolean hypercube under a mapping
, where each coordinate is a constant-degree
multilinear polynomial with 0-1 coefficients. We show the following
applications of our bounds. Let be the random
subset of containing each element independently with
probability .
A set is -intersective if
any dense subset of contains a proper -term
arithmetic progression with common difference in . Our main result implies
that is -intersective with probability provided for . This gives a polynomial improvement for all
of a previous bound due to Frantzikinakis, Lesigne and Wierdl, and
reproves more directly the same improvement shown recently by the authors and
Dvir.
Let be the number of -term arithmetic progressions in
and consider the large deviation rate
. We give quadratic
improvements of the best-known range of for which a highly precise estimate
of due to Bhattacharya, Ganguly, Shao and Zhao is valid for
all odd .
We also discuss connections with error correcting codes (locally decodable
codes) and the Banach-space notion of type for injective tensor products of
-spaces.Comment: 18 pages, some typos fixe
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
Gaussian width bounds with applications to arithmetic progressions in random settings
Motivated by two problems on arithmetic progressions (APs)—concerning large
deviations for AP counts in random sets and random differences in Szemer´edi’s theorem—
we prove upper bounds on the Gaussian width of the image of the n-dimensional Boolean
hypercube under a mapping ψ : Rn → Rk, where each coordinate is a constant-degree
multilinear polynomial with 0/1 coefficients. We show the following applications of our
bounds. Let [Z/NZ]p be the random subset of Z/NZ containing each element independently
with probability p.
• Let Xk be the number of k-term APs in [Z/NZ]p. We show that a precise estimate
on the large deviation rate log Pr[Xk ≥ (1 + δ)EXk] due to Bhattacharya, Ganguly,
Shao and Zhao is valid if
On Embeddings of l_1^k from Locally Decodable Codes
We show that any -query locally decodable code (LDC) gives a copy of
with small distortion in the Banach space of -linear forms on
, provided and where , , and the distortion are simple functions of the code
parameters. We exhibit the copy of by constructing a basis for it
directly from "smooth" LDC decoders. Based on this, we give alternative proofs
for known lower bounds on the length of 2-query LDCs. Using similar techniques,
we reprove known lower bounds for larger . We also discuss the relation with
an alternative proof, due to Pisier, of a result of Naor, Regev, and the author
on cotype properties of projective tensor products of spaces
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Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term “outlaw distributions” for such distributions since they “defy” the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently “smooth” functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters