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Dmitry Faifman

Publications and source records attributed to Dmitry Faifman.

24 records · Page 2Linked to original sources
arXiv2014

On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus

Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant function. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. We show that when $k \leq c \log n / (\log \log n + \log 1/\varepsilon)$, the maximum and the minimum of $f$ on this random subtorus $M$ differ by at most $\varepsilon$, with high probability.

math.FA↗
arXiv2011

The quotient girth of normed spaces, and an extension of Schäffer's dual girth conjecture to Grassmannians

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove the analogs of Schäffer's dual girth conjecture (proved by Álvarez-Paiva) and the Holmes-Thompson dual volumes theorem in the projective setting. We then show that the projective Finsler structure admits a natural extension to higher Grassmannians, and prove the corresponding theorems in the general case. We follow Álvarez-Paiva's approach to the problem, namely, we study the symplectic geometry of the associated co-ball bundles. For the higher Grassmannians, the theory of Hamiltonian actions is applied.

math.MG↗
arXiv2011

On Multiplicative Maps of Continuous and Smooth Functions

In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.

math.CA↗
arXiv2011

A Family of Unitary Operators Satisfying a Poisson-type Summation Formula

We consider a weighted form of the Poisson summation formula. We prove that under certain decay rate conditions on the weights, there exists a unique unitary Fourier-Poisson operator which satisfies this formula. We next find the diagonal form of this operator, and prove that under weaker conditions on the weights, a unique unitary operator still exists which satisfies a Poisson summation formula in operator form. We also generalize the interplay between the Fourier transform and derivative to those Fourier-Poisson operators.

math.CA↗
arXiv2008

Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.

math.NT↗