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Growth of the product j = 1 n ( 1 - x a j )

J. P. BellP. B. BorweinL. B. Richmond — 1998

Acta Arithmetica

We estimate the maximum of j = 1 n | 1 - x a j | on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when a j is j k or when a j is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when a j is j.    In contrast we show, under fairly general conditions, that the maximum is less than 2 n / n r , where r is an arbitrary positive number. One consequence is that the...

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