87,414 research outputs found
On the sum of the L1 influences of bounded functions
Let have degree as a multilinear
polynomial. It is well-known that the total influence of is at most .
Aaronson and Ambainis asked whether the total influence of can also
be bounded as a function of . Ba\v{c}kurs and Bavarian answered this
question in the affirmative, providing a bound of for general
functions and for homogeneous functions. We improve on their results
by providing a bound of for general functions and for
homogeneous functions. In addition, we prove a bound of for
monotone functions, and provide a matching example.Comment: 16 pages; accepted for publication in the Israel Journal of
Mathematic
Strong Contraction and Influences in Tail Spaces
We study contraction under a Markov semi-group and influence bounds for
functions in tail spaces, i.e. functions all of whose low level Fourier
coefficients vanish. It is natural to expect that certain analytic inequalities
are stronger for such functions than for general functions in . In the
positive direction we prove an Poincar\'{e} inequality and moment decay
estimates for mean functions and for all , proving the degree
one case of a conjecture of Mendel and Naor as well as the general degree case
of the conjecture when restricted to Boolean functions. In the negative
direction, we answer negatively two questions of Hatami and Kalai concerning
extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. That
is, we construct a function whose Fourier
coefficients vanish up to level , with all influences bounded by for some constants . We also construct a function
with nonzero mean whose remaining Fourier
coefficients vanish up to level , with the sum of the influences
bounded by for some constants
.Comment: 20 pages, two new proofs added of the main theore
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