Professor Miklos Laczkovich
(Jerusalem/Hungarian Academy of Sciences)
"The difference property"
Abstract:
Let G_1 and G_2 be Abelian groups, and let Delta_h denote the difference operator:
Delta _h f (x) = f(x+h)-f(x)
for every f:G_1 -->G_2 and x, h in G_1. We say that a class F of functions mapping G_1 into G_2 has the difference property if, whenever f:G_1 --> G_2 is such that Delta_h f lies in F for every h in G_1, then f = g+H, for some g in F and homomorphism H from G_1 into G_2 .This notion was introduced by De Bruijn in 1951. He established the difference property of the class of continuous real functions (a conjecture of Erdos) and that of several other classes, including C^k(R), C^infty(R), the classes analytic functions, of polynomials, and many others.
De Bruijn's results and methods were generalized in many directions, and during the past 50 years dozens of papers were written in the subject. We give a survey of the topic and also indicate its connection with some other areas, e.g. with stability theorems, small sets of harmonic analysis or, in connection with the difference property of the class of measurable functions, with some independence results on cardinal invariants of ideals.
Coffee, Cookies at the faculty lounge at 3:30.