Abstract
We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method. This equivalence is used to give simple proofs of theorems of J. Brüdern.
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References
J. Brüdern, Binary additive problems and the circle method, multiplicative sequences and convergent sieves, to appear.
J. Brüdern, A. Granville, A. Perelli, R. C. Vaughan and T. D. Wooley, On the exponential sum over k-free numbers, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 356 (1998), 739–761.
W. Schwarz and J. Spilker, Arithmetical functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties. London Mathematical Society Lecture Note Series. 184 (Cambridge, 1994).
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Puchta, JC. An additive property of almost periodic sets. Acta Mathematica Hungarica 97, 323–331 (2002). https://doi.org/10.1023/A:1021613315705
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DOI: https://doi.org/10.1023/A:1021613315705