Glasnik Matematicki, Vol. 45, No.2 (2010), 357-372.

ON VAN DER CORPUT PROPERTY OF SQUARES

Siniša Slijepčević

Department of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
e-mail: slijepce@math.hr

Abstract.   We prove that the upper bound for the van der Corput property of the set of perfect squares is O((log n)-1/3), giving an answer to a problem considered by Ruzsa and Montgomery. We do it by constructing non-negative valued, normed trigonometric polynomials with spectrum in the set of perfect squares not exceeding n, and a small free coefficient a0=O((log n)-1/3).

2000 Mathematics Subject Classification.   11P99, 37A45.

Key words and phrases.   Sárközy theorem, recurrence, difference sets, positive definiteness, van der Corput property, Fourier analysis.

Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.05

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