Abstract
Using the Rudin-Shapiro sequence, the existence of a regular but not strongly regular positive matrix that sums { exp ( 2 π i k t ) } \{ \exp (2\pi ikt)\} to 0 for all t ∈ ( 0 , 1 ) t \in (0,1) is established. As a corollary it is shown that there exist matrices that sum all almost periodic sequences without possessing the Borel property and vice versa.
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