Rad HAZU, Matematičke znanosti, Vol. 18 (2014), 35-53.

POSITIVE EXPONENTIAL SUMS AND ODD POLYNOMIALS

Marina Ninčević and Siniša Slijepčević

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

Abstract.   Given an odd integer polynomial f(x) of a degree k ≥ 3, we construct a non-negative valued, normed trigonometric polynomial with non-vanishing coefficients only at values of f(x) not greater than n, and a small free coefficient a0 = O((log n)-1/k). This gives an alternative proof of the bound for the maximal possible cardinality of a set of integers A, so that A - A does not contain an integer value of f(x). We also discuss other interpretations and an ergodic characterization of that bound.

2010 Mathematics Subject Classification.   11P99, 37A45.

Key words and phrases.   Positive exponential sums, van der Corput sets, correlative sets, recurrence, difference sets, Fejér’s kernel, positive definiteness.

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