Jason Rute

Algorithmic randomness, reverse mathematics, and the dominated convergence theorem

Abstract We analyze the pointwise convergence of a sequence of computable elements of L 1 ( 2 ω ) in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory R C A 0 , each is equivalent to the assertion that every G δ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Konigʼs lemma relativized to the Turing jump of any set. It is also equivalent to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ 2 collection.

Oscillation and the mean ergodic theorem for uniformly convex Banach spaces

Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p, where the constant C depends only on p and the modulus of uniform convexity. For T a nonexpansive operator, we obtain a weaker bound on the number of epsilon-fluctuations in the sequence. We clarify the relationship between bounds on the number of epsilon-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.

A metastable dominated convergence theorem

The dominated convergence theorem implies that if (fn) is a se- quence of functions on a probability space taking values in the interval (0,1), and (fn) converges pointwise a.e., then ( R fn) converges to the integral of the pointwise limit. Tao (20) has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypoth- esis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space. We prove a slight strengthening of Taos theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorovs theorem, and introduce a new mode of convergence related to these notions.

Algorithmic Randomness and Fourier Analysis

Suppose 1 < p < ∞. Carleson’s Theorem states that the Fourier series of any function in Lp[−π, π] converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every f ∈ Lp[−π, π] given natural computability conditions on f and p.

Schnorr randomness for noncomputable measures

This paper explores a novel definition of Schnorr randomness for noncomputable measures. We say

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

x

A FORMAL PROOF OF THE KEPLER CONJECTURE

is uniformly Schnorr

Van Lambalgen's theorem for uniformly relative Schnorr and computable randomness

\mu

Computable randomness and betting for computable probability spaces.

-random if

Topics in algorithmic randomness and computable analysis

t(\mu,x)<\infty