The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where $\mathcal{Z}$ is the $\mathbb{Q}$-span of the set of multiple zeta values (of positive integer arguments). My picture of mixed Tate motives is not very clear, and I would like to be able to relate their periods to something I understand better.

There is a survey article of Kontsevich and Zagier which defines a period as a complex number whose real and imaginary parts are given by convergent integrals of rational functions with rational coefficients, over domains in $\mathbb{R}^n$ cut out by finitely many polynomial inequalities with rational coefficients.

What is the relationship between the set of periods of mixed Tate motives over $\mathbb{Z}$ and the set of periods in the sense of Kontsevich/Zagier? Does one of these sets contain the other?

I would be interested to see examples of periods of one kind which are not periods of the other.

thisis the point. $\endgroup$ – user9072 Apr 7 '13 at 21:13