arXiv subjects
Publications and source records attributed to Gady Kozma.
We show that for every finite symetric set S of integer vectors, every real trigonometric polynomial on the d dimensional torus with spectrum in S has a zero in every closed ball of diameter D, where D is the sum over S of 1 over 4 times the L2 norm of the vector. We investigate tightness in some special cases.
Excited random walk is a process that has a drift to the right whenever it encounters a new vertex. The paper shows that in two dimensions it drifts to the right linearly in time.
Inspired by Menshov's representation theorem, we prove that there exists a sequence of frequecies such that any measurable (complex valued) function on R can be represented as a sum of almost everywhere convergent trigonometric series with these frequencies.
Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might get a Menshov type representation converging almost everywhere or in measure.
Let phi be a Dubins-Freedman random homeomorphism on [0,1] derived from the base measure uniform on the vertical line x=1/2, and let f be a periodic function satisfying that |f(x)-f(0)| = o(1/log log log 1/x). Then the Fourier expansion of f composed with phi converges at 0 with probability 1. In the condition on f, o cannot be replaced by O. Also we deduce some 0-1 laws for this kind of problems.
A Menshov spectrum is a subset of the integers that is sufficient for representing every measurable function as an almost-everywhere converging trigonometric (non-Fourier) sum. In this language the celebrated "Menshov representation theorem" states that Z is a Menshov spectrum. In this paper we construct 1) Menshov spectra that are almost exponentially sparse 2) that are almost squares. Then we show that the positive integers are not a Menshov spectrum but are a Menshov spectrum in measure, and repeat 1) and 2) in the analytic settings.
We show that a spectrum of frequencies obtained by a random perturbation of the integers allows one to represent any measurable function on R by an almost everywhere converging sum of harmonics almost surely.
We proved recently math.CA/0510403 that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may be square integrable on the circle. Here we prove that it can even be infinitely differentiable, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a "new quasi-analyticity".
We show that it is possible for a square integrable function on the circle, which is a sum of an almost everywhere convergent series of exponentials with positive frequencies, to not belong to the Hardy space. A consequence in the uniqueness theory is obtained.
We show that the scaling limit exists and is invariant to dilations and rotations. We give some tools that might be useful to show universality.
We examine the class of functions representable by an analytic sum (by which we mean a trigonometric sum involving only positive frequencies) converging almost everywhere. We show that it is dense but that it is first category and has zero Wiener measure.
If B is a compact space and B\{pt} is Lindelof then B^k\{pt} is star-Linedlof for every cardinality k. If B\{pt} is compact then B^k\{pt} is discretely star-Lindelof. In particular, this gives new examples of Tychonoff discretely star-Lindelof spaces with unlimited extent.
We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$.
We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.
We show that the alpha-weight of an MST over n points in a metric space with upper box dimension d has a bound independent of n if alpha is smaller than d and does not have one if alpha is larger than d.
We observe returns of a simple random walk on a finite graph to a fixed node, and would like to infer properties of the graph, in particular properties of the spectrum of the transition matrix. This is not possible in general, but at least the eigenvalues can be recovered under fairly general conditions, e.g. when the graph has a node-transitive automorphism group. The main result is that by observing polynomially many returns, it is possible to estimate the spectral gap of such a graph up to a constant factor.
Excited random walk is a random walk that has a positive drift to the right when it reaches a vertex it hasn't been to before. We show that in three dimensions the walk drifts to the right in non-zero speed.
Sharp estimates for the length of loop erased random walk between two vertices on the [n]^d -torus, d > 4, are established. The mean length is order n^{d/2} . In dimension 4 we have only an upper bound.