Related papers: Effective Randomness for Continuous Measures
Random matrices like GUE, GOE and GSE have been studied for decades and have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbj{\o}rnsen in…
We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable…
Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P.
We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show…
There is an infinite subset of a Martin-L\"of random set of integers that does not compute any Martin-L\"of random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset…
We investigate the asymptotic densities of theorems provable in Zermelo-Fraenkel set theory ZF and its extension ZFC including the axiom of choice. Assuming a canonical De Bruijn representation of formulae, we construct asymptotically large…
We prove consistency of the following sentence: ``ZFC + every real function is continuous on a non-meagre set'', answering a question of Fremlin.
We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…
This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set…
We study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0, 2, 4, 6, 8\}$. Computational checks up to $n = 10^{15}$ reveal the known cases $n = 1, 2, 3, 6, 11$ and no…
We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element extendable to a path, a given countable class $\mathcal{M}$ closed under disjoint union, and any set $A$, if none of the members of $\mathcal{M}$ strongly…
Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a…
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable?…
In the first part of this paper the notion of natural metric on the set of natural numbers is defined. It is such metric that the completion of N is a compact metric space that a probability borel measure exists in order that the sequence…
The intrinsic random nature of quantum physics offers novel tools for the generation of random numbers, a central challenge for a plethora of fields. Bell non-local correlations obtained by measurements on entangled states allow for the…
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by…
We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We…
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions…
In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally,…
Consider the ring $C_c(X)_F$ of real valued functions which are discontinuous on a finite set with countable range. We discuss $(\mathcal{Z}_c)_F$-filters on $X$ and $(\mathcal{Z}_c)_F$-ideals of $C_c(X)_F$. We establish an analogous…