arXiv subjects

Patrick LaVictoire

Publications and source records attributed to Patrick LaVictoire.

7 recordsLinked to original sources

arXiv2021

Robust Cooperation in the Prisoner's Dilemma: Program Equilibrium via Provability Logic

We consider the one-shot Prisoner's Dilemma between algorithms with read-access to one anothers' source codes, and we use the modal logic of provability to build agents that can achieve mutual cooperation in a manner that is robust, in that cooperation does not require exact equality of the agents' source code, and unexploitable, meaning that such an agent never cooperates when its opponent defects. We construct a general framework for such "modal agents", and study their properties.

cs.GT↗

arXiv2012

Maximal Functions for Lacunary Dilation Structures

If mu is a smooth density on a hypersurface in R^d whose curvature never vanishes to infinite order, and A is a d-by-d matrix whose eigenvalues all have absolute value greater than 1, then the maximal function given by convolving f with dilates of mu by powers of A, and taking the maximum, is bounded from a corresponding version of H^1 to weak L^1.

math.CA↗

arXiv2012

Multivariable averaging on sparse sets

Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christ's result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a "native" d-dimensional average and a "product" average from the 1-dimensional averages.

math.DS↗

arXiv2011

Pointwise Convergence for Subsequences of Weighted Averages

We prove that if $μ_n$ are probability measures on $Z$ such that $\hat μ_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $μ_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor ρ(n)\rfloor$ for a slowly growing function $ρ$. Under some monotonicity assumptions, the rate of growth of $ρ'(x)$ determines the existence of a "good" subsequence of these averages.

math.CA↗

arXiv2009

Universally L^1-Bad Arithmetic Sequences

We present a modified version of Buczolich and Mauldin's proof that the sequence of square numbers is universally L^1-bad. We extend this result to a large class of sequences, including the dth powers and the set of primes; furthermore, we show that any subsequence of the averages taken along these sequences is also universally L^1-bad.

math.DS↗

arXiv2008

L^1 Ergodic Theorems for Random Group Averages

This is an earlier, but more general, version of "An L^1 Ergodic Theorem for Sparse Random Subsequences". We prove an L^1 ergodic theorem for averages defined by independent random selector variables, in a setting of general measure-preserving group actions. A far more readable version of this paper is in the works.

math.DS↗

arXiv2008

An L^1 Ergodic Theorem for Sparse Random Subsequences

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \ell^1 convolution operators.

math.DS↗