# Jerusalem Mathematics Colloquium

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Thursday, 30th November 2000, 4:00 pm

Mathematics Bldg., lecture hall 2

##

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.

You are invited to join the speaker for further discussion after the
talk at Beit Belgia.

List of talks, 2000-01

List of talks, 1998-99

List of talks, 1997-98