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Revision #2 to TR09-004 | 19th May 2009 00:00

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

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Abstract:

We extend the ``method of multiplicities''
to get the following
results, of interest in combinatorics and randomness extraction.
\begin{enumerate}
\item
We show that every Kakeya set (a set of points that
contains a line in every direction) in $\F_q^n$
must be of size at least $q^n/2^n$.
This bound is tight to within a $2 + o(1)$ factor for
every $n$ as $q \to \infty$, compared to previous
bounds that were off by exponential factors in $n$.
\item
We give improved randomness extractors and ``randomness mergers''.
Mergers are seeded functions that take as input $\Lambda$ (possibly
correlated) random variables in $\{0,1\}^N$ and a short random
seed and output a single random variable in $\{0,1\}^N$ that is
statistically close to having entropy $(1-\delta) \cdot N$ when
one of the $\Lambda$ input variables is distributed uniformly. The
seed we require is only $(1/\delta)\cdot \log \Lambda$-bits long,
which significantly improves upon previous construction of
mergers. \item Using our new mergers, we show how to construct
randomness extractors that use logarithmic length seeds while
extracting $1 - o(1)$ fraction of the min-entropy of the source.
Previous results could extract only a constant fraction of the
entropy while maintaining logarithmic seed length.
\end{enumerate}
The ``method of multiplicities", as used in prior work,
analyzed subsets of vector spaces over
finite fields by constructing somewhat low degree
interpolating
polynomials that vanish on every point in the subset {\em
with high multiplicity}.
The typical use of this method involved showing that the
interpolating polynomial also vanished on some points outside
the subset, and then used simple bounds on the number
of zeroes to complete the analysis. Our augmentation to this technique
is that we prove, under appropriate conditions,
that the interpolating polynomial vanishes {\em with high multiplicity}
outside the set. This novelty leads to significantly tighter analyses.

To develop the extended method of multiplicities we provide a number
of basic technical results
about multiplicity of zeroes of polynomials that may be of general
use. For instance, we strengthen the
Schwartz-Zippel lemma to show that the expected multiplicity
of zeroes of a non-zero degree $d$ polynomial at a random point in
$S^n$, for any finite subset $S$ of the underlying field, is at most
$d/|S|$ (a fact that does not seem to have been noticed in the
CS literature before).


Revision #1 to TR09-004 | 16th January 2009 00:00

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers





Revision #1
Authors: Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan
Accepted on: 16th January 2009 00:00
Downloads: 1287
Keywords: 


Abstract:

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction. begin{enumerate} item We show that every Kakeya set in $F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q to infty$. item We give improved ``randomness mergers'', i.e., seeded functions that take as input $k$ (possibly correlated) random variables in ${0,1}^N$ and a short random seed and output a single random variable in ${0,1}^N$ that is statistically close to having entropy $(1-delta) cdot N$ when one of the $k$ input variables is distributed uniformly. The seed we require is only $(1/delta)cdot log k$-bits long, which significantly improves upon previous construction of mergers. end{enumerate} The ``method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses. To get the extended method of multiplicities we provide a number of basic technical results about multiplicity of zeroes of polynomials that may be of general use. For instance, we strengthen the Schwartz-Zippel lemma to show that the expected multiplicity of zeroes of a non-zero degree $d$ polynomial at a random point in $S^n$, for any finite subset $S$ of the underlying field, is at most $d/|S|$ (a fact that does not seem to have been noticed in the CS literature before).


Paper:

TR09-004 | 15th January 2009 00:00

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers





TR09-004
Authors: Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan
Publication: 15th January 2009 14:51
Downloads: 1189
Keywords: 


Abstract:

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction.
\begin{enumerate}
\item We show that every Kakeya set in $\F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q \to \infty$.
\item We give improved ``randomness mergers'', i.e., seeded functions that take as input $k$ (possibly correlated) random variables in $\{0,1\}^N$ and a short random seed and output a single random variable in $\{0,1\}^N$ that is statistically close to having entropy $(1-\delta) \cdot N$ when one of the $k$ input variables is distributed uniformly. The seed we require is only $(1/\delta)\cdot \log k$-bits long, which significantly improves upon previous construction of mergers.
\end{enumerate}

The ``method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {\em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {\em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses.

To get the extended method of multiplicities we provide a number of basic technical results about multiplicity of zeroes of polynomials that may be of general use. For instance, we strengthen the Schwartz-Zippel lemma to show that the expected multiplicity of zeroes of a non-zero degree $d$ polynomial at a random point in $S^n$, for any finite subset $S$ of the underlying field, is at most $d/|S|$ (a fact that does not seem to have been noticed in the CS literature before).



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