Related papers: Resource bounded Ku\v{c}era-G\'{a}cs Theorems

A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and…

This article describes a Turing machine which can solve for $\beta^{'}$ which is RE-complete. RE-complete problems are proven to be undecidable by Turing's accepted proof on the Entscheidungsproblem. Thus, constructing a machine which…

Limitation of computational resources is considered as a universal principle that for simulation is as fundamental as physical laws are. It claims that all experimentally verifiable implications of physical laws can be simulated by the…

We establish rigorous error bounds for prime counting using a truncated Gaussian (TG) kernel in the explicit formula framework. Our main theorem proves that the approximation error remains globally below 1/2 for all sufficiently large…

An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so; it…

This paper presents finite-blocklength achievability bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a…

A Random Access query to a string $T\in [0..\sigma)^n$ asks for the character $T[i]$ at a given position $i\in [0..n)$. In $O(n\log\sigma)$ bits of space, this fundamental task admits constant-time queries. While this is optimal in the…

It is known that distillation in continuous variable resource theories is impossible when restricted to Gaussian states and operations. To overcome this limitation, we enlarge the theories to include convex mixtures of Gaussian states and…

We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural ``Ville event'' $\mathcal E_\alpha$, this regret till time $T$…

Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of…

We prove that, for every theory $T$ which is given by an ${\mathcal L}_{\omega_1,\omega}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^\omega$ on a cone of Turing degrees,…

We study quasi-stationary distributions and quasi-limiting behavior of Markov chains in general reducible state spaces with absorption. We propose a set of assumptions dealing with particular situations where the state space can be…

In this paper, we study the Schr\"odinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables…

Infinite time Turing machine models with tape length $\alpha$, denoted $T_\alpha$, strengthen the machines of Hamkins and Kidder [HL00] with tape length $\omega$. A new phenomenon is that for some countable ordinals $\alpha$, some cells…

One of the earliest models of weak randomness is the Chor-Goldreich (CG) source. A $(t,n,k)$-CG source is a sequence of random variables $X=(X_1,\dots,X_t)\sim(\{0,1\}^n)^t$, where each $X_i$ has min-entropy $k$ conditioned on any fixing of…

Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For \epsilon > 0, we say that x_i is an \epsilon-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less…

We propose a new class of fast Krasnoselkii--Mann methods with variance reduction to solve a finite-sum co-coercive equation $Gx = 0$. Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators…

We consider the quantum decoding problem. It consists in recovering a codeword given a superposition of noisy versions of this codeword. By measuring the superposition, we get back to the classical decoding problem. It appears for the first…

Let X be randomly chosen from {-1,1}^n, and let Y be randomly chosen from the standard spherical Gaussian on R^n. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr [X belongs to P] - Pr [Y…

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the…