**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor Walter Rudin, Department of Mathematics,
University of Wisconsin-Madison

*Analytic aspects of the Bohr Topology*

If G is an abelian group, then G^# denotes G equipped with the
weakest topology that makes every character of G continuous.
This is the Bohr topology of G. If G= \Bbb Z, the additive group
of the integers, and A is a Hadamard set in \Bbb Z, it is shown that:
(i) A-A has 0 as its only limit point in \Bbb Z^#, (ii) No Sidon
subset of A-A has a limit point in \Bbb Z^#, (iii) A-A is a
\Lambda (p) set for all p<\infty. This leads to an explicit example
of a set which is \Lambda (p) for all p<\infty and is dense in
\Bbb Z^#. If f(x) is a quadratic or cubic polynomial with integer
coefficients, then the closure of f(\Bbb Z) in the Bohr
compactification of \Bbb Z is shown to have Haar measure 0.

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