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Register a new userA Three Dimensional Signed Small Ball Inequality
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The Small Ball Inequality is a conjectural lower bound on sums the L-infinity norm of sums of Haar functions supported on dyadic rectangles of a fixed volume in the unit cube. The conjecture is fundamental to questions in discrepancy theory, approximation theory and probability theory. In this article, we concentrate on a special case of the conjecture, and give the best known lower bound in dimension 3, using a conditional expectation argument.
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This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions. In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved.
Let h_R denote an L ^{infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{infty} norm of the `hyperbolic sums $$ sum _{|R|=2 ^{-n}} alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{eta} for some positive eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3.
A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory.
Consider the trilinear form for twisted convolution on $mathbb{R}^{2d}$: begin{equation*} mathcal{T}_t(mathbf{f}):=iint f_1(x)f_2(y)f_3(x+y)e^{itsigma(x,y)}dxdy,end{equation*} where $sigma$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t eq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $sigma$.
We show how Turans inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality $|x|P_n(x)^2-P_{n-1}(x)P_{n+1}(x)geq 0$, $-1leq xleq 1$, effortlessly with the aid of our method.
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