Related papers: Higher Kurtz randomness

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-L\"{o}f random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly…

For an irreducible variety $X$ over a field $k$, the degree of irrationality $\operatorname{irr}_k X$ is the minimal degree of a dominant rational map $X \dashrightarrow \mathbb{P}_k^{\operatorname{\dim} X}$. When $X$ is a curve, this is…

We characterize the topological non-cancellative cones that are expressible as projective limits of finite powers of $[0,\infty]$. These are also the cones of lower semicontinuous extended-valued traces on AF C*-algebras. Our main result…

We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a…

Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…

We establish character rigidity for all non-uniform higher-rank irreducible lattices in semisimple groups of characteristic other than 2. This implies stabilizer rigidity for probability measure preserving actions and rigidity of invariant…

The one-dimensional isotropic quantum Heisenberg spin systems with random couplings and random spin sizes are investigated using a real-space renormalization group scheme. It is demonstrated that these systems belong to a universality class…

Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying…

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chosen with respect to this measure is discrete with probability…

Let At denote the set of infinite sequences of effective dimension t. We determine both how close and how far an infinite sequence of dimension s can be from one of dimension t, measured using the Besicovitch pseudometric. We also identify…

There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.

We prove that the set of closed orbits in a real reductive representation contains a subset which is open with respect to the real Zariski topology if it has non-empty interior. In particular the set of closed orbits is dense.

We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and…

Let $E$ be a bounded open subset of $\mathbb{R}^n$. We study the following questions: For i.i.d. samples $X_1, \dots, X_N$ drawn uniformly from $E$, what is the probability that $\cup_i \mathbf{B}(X_i, \delta)$, the union of $\delta$-balls…

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Delta^0_{n+1} basic sequence with respect to which no…

Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity…

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies…

We investigate the properties of a class of piecewise-fractional maps arising from the introduction of an invariance under rescaling into convex quadratic maps. The subsequent maps are quasiconvex, and pseudoconvex on specific convex cones;…

A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed $p$ (characteristic) and $d$ (dimension), there exist a…