The Journal of Fourier Analysis and Applications Volume4, Issue 2, 1998 Applications of Generalized Perron Trees to Maximal Functions and Density Bases Kathryn E. Hare and Jan-Olav ROnning Communicated by Fernando Soria ABSTRAC'E In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate L P-functions or give Hardy- Littlewood type maximal functions which are bounded on L p, p > 1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et aL [61 regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above. 1. Introduction An important problem in harmonic analysis is the question of the differentiability of integrals in R 2, or more generally R n. One formulation of this problem is the following: Consider a set A C T, the unit circle in R 2, and view A as a selection of directions. If Ax is the collection of all rectangles in R 2 containing x and oriented in one of the directions in A, is it true that for "all" f diamR--~olim ,--,1 fR f(y) dy = f(x) a.e. ? REAx If this is true for some class of functions, we say that A differentiates that class of functions. Closely related to this is the problem of the LP-boundedness of the corresponding Hardy-Littlewood maximal operator MA, defined by MA(f) = sup If(Y)[dY. REAx -~1 Math Subject Classifications. Primary: 42B25. Keywords and Phrases. Maximal functions, density basis, Perron trees, lacunary sets. Acknowledgements andNotes. This research was partially supported by NSERC. 1998 Birkh~iuser Boston. All tights reserved ISSN 1069-5869