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# Documents  37A45 | enregistrements trouvés : 3

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## Post-edited  Integral points on Markoff type cubic surfaces and dynamics Sarnak, Peter (Auteur de la Conférence) | CIRM (Editeur )

Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space. Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the ...

## Multi angle  Ergodicity of the Liouville system implies the Chowla conjecture Frantzikinakis, Nikos (Auteur de la Conférence) | CIRM (Editeur )

The Chowla conjecture asserts that the signs of the Liouville function are distributed randomly on the integers. Reinterpreted in the language of ergodic theory this conjecture asserts that the Liouville dynamical system is a Bernoulli system. We prove that ergodicity of the Liouville system implies the Chowla conjecture. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary decomposition results involving uniformity norms, and equidistribution results on nilmanifolds. The Chowla conjecture asserts that the signs of the Liouville function are distributed randomly on the integers. Reinterpreted in the language of ergodic theory this conjecture asserts that the Liouville dynamical system is a Bernoulli system. We prove that ergodicity of the Liouville system implies the Chowla conjecture. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary ...

## Multi angle  Möbius randomness and dynamics six years later Sarnak, Peter (Auteur de la Conférence) | CIRM (Editeur )

There have many developments on the disjointness conjecture of the Möbius (and related) function to topologically deterministic sequences. We review some of these highlighting some related arithmetical questions.

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