220 research outputs found

### Uncomputably noisy ergodic limits

V'yugin has shown that there are a computable shift-invariant measure on
Cantor space and a simple function f such that there is no computable bound on
the rate of convergence of the ergodic averages A_n f. Here it is shown that in
fact one can construct an example with the property that there is no computable
bound on the complexity of the limit; that is, there is no computable bound on
how complex a simple function needs to be to approximate the limit to within a
given epsilon

### Uniform distribution and algorithmic randomness

A seminal theorem due to Weyl states that if (a_n) is any sequence of
distinct integers, then, for almost every real number x, the sequence (a_n x)
is uniformly distributed modulo one. In particular, for almost every x in the
unit interval, the sequence (a_n x) is uniformly distributed modulo one for
every computable sequence (a_n) of distinct integers. Call such an x "UD
random". Here it is shown that every Schnorr random real is UD random, but
there are Kurtz random reals that are not UD random. On the other hand, Weyl's
theorem still holds relative to a particular effectively closed null set, so
there are UD random reals that are not Kurtz random

### Mathematics and language

This essay considers the special character of mathematical reasoning, and
draws on observations from interactive theorem proving and the history of
mathematics to clarify the nature of formal and informal mathematical language.
It proposes that we view mathematics as a system of conventions and norms that
is designed to help us make sense of the world and reason efficiently. Like any
designed system, it can perform well or poorly, and the philosophy of
mathematics has a role to play in helping us understand the general principles
by which it serves its purposes well

### Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page

### Ultraproducts and metastability

Given a convergence theorem in analysis, under very general conditions a
model-theoretic compactness argument implies that there is a uniform bound on
the rate of metastability. We illustrate with three examples from ergodic
theory

### The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression

In 1837, Dirichlet proved that there are infinitely many primes in any
arithmetic progression in which the terms do not all share a common factor. We
survey implicit and explicit uses of Dirichlet characters in presentations of
Dirichlet's proof in the nineteenth and early twentieth centuries, with an eye
towards understanding some of the pragmatic pressures that shaped the evolution
of modern mathematical method

### Quantifier elimination for the reals with a predicate for the powers of two

In 1985, van den Dries showed that the theory of the reals with a predicate
for the integer powers of two admits quantifier elimination in an expanded
language, and is hence decidable. He gave a model-theoretic argument, which
provides no apparent bounds on the complexity of a decision procedure. We
provide a syntactic argument that yields a procedure that is primitive
recursive, although not elementary. In particular, we show that it is possible
to eliminate a single block of existential quantifiers in time $2^0_{O(n)}$,
where $n$ is the length of the input formula and $2_k^x$ denotes $k$-fold
iterated exponentiation

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