The Department of Analysis of Eötvös Loránd University,

The Department of Analysis of Eötvös Loránd University, PRESENTS in cooperation with Central European University, and Limage Holding SA

Balcerzak Functions... Méla Differences... Host ...and their differences Tamás Mátrai Kahane Keleti Buczolich Parreau Imre Ruzsa Miklós Laczkovich

”If f is a measurable real function such that the difference functions f(x+h) - f(x) are continuous for every real h, for every real h, then f itself is continuous.” How many h’s should we consider?

If B and S are two classes of real functions onTwith S B then H(B,S)= H T : there is an fB \S such that h f S for every h H T :circle group h f = f(x+h) - f(x)

fis measurable, h fcontinuous for everyhT T H(B,S) fis continuous B -measurable functions S -continuous functions Example on T :

B: L1 (T) S:L2(T) Work schedule: • H(B,S) for special function classes; • translation to general classes (simple) • done!

aie2πint f ~ H H(L1,L2) H ||h f|| < 1 , h L2 ai(e2πin(t+h)- e2πint) = h f = dµ(h) dµ(h) ai e2πint(e2πinh -1) dµ(h) measure concentrated on H (e2πinh -1) >  > 0? dµ(h) What if Upper bound for H(L1,L2):

T Borel set H is weak Dirichlet if for every probability measure µ concentrated on H, (e2πinh -1) =0 dµ(h) H(L1,L2) H(L1,L2) weak Dirichlet sets weak Dirichlet sets Weak Dirichlet sets:

L1\L2: Wanted f L2 h f for every T H symetric difference h H A T A , f = h f = f(x+h)-f(x) = =A(x+h)-A(x)=A∆(A+h) (A∆(A+h)) is very small for every h H? What if (A) is big, while Lower bound for H(L1,L2): Try characteristic functions! Lebesgue measure

is nonejectiveiff there is a  > 0: T H (A∆(A+h))=0 Nonejective sets H(L1,L2) Nonejective sets H(L1,L2) Nonejective sets:

H(L1,L2) Every is a subset of an F subgroup ofT. H T. Keleti: sets of absolute convregence of not everywhere convergent Fourier series is anN-set iff it can be T T Host Méla Parreau H H : covered by a countable union of weak Dirichlet sets Compact is weak Dirichlet iff I. Ruzsa: it is nonejective. Some lemmas: H(L1,L2) =N - sets H(L1,L2) =N - sets

“A set is as ejective as far from being Weak Dirichlet.” F = 1, ||f||L ={f L2: } f = 0 2 T M (H) = {probability measures on H} |e2inh-1|2 dµ(h) T 2 ||∆hf||L 2 Moreover: =

Lp Lp f f    L L  hf hf if  >1 p Only for 0 q 2: Translation for other classes: Take powers: H(Lp,Lq) =N - sets H(Lp,Lq) =N - sets

H(Lp,ACF)= N , 0<p< , 0<p< H(Lp,L )=F H(Lip,Lip) , 0<<<1, classes coincide Some other classes (T. Keleti): END very complicated H(B,C)

The Department of Analysis of Eötvös Loránd University,