A structure theorem for multiplicative functions over the Gaussian integers and applications
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We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2−y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open.
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- V. Bergelson, Ergodic Theory and Diophantine Problems: Topics in Symbolic Dynamics and Applications, Cambridge Univ. Press, Cambridge, 1996, pp. 167–205.Google Scholar
- N. Frantzikinakis and B. Host, Uniformity of multiplicative functions and partition regularity of some quadratic equations, arXiv: 1303. 4329.Google Scholar
- G. H. Hardy and S. Ramanujan, Twelve Lectures on Subjects Suggested by his Life and Work, 3rd ed., Chelsea, New York, 1999, p. 67.Google Scholar
- B. Szegedy, On higher order Fourier analysis, arXiv: 1203. 2260.Google Scholar
- P. Tchebichef, Mémoire sur les nombres premiers, J. Math. Pures Appl. 17 (1852), 366–390.Google Scholar