Acta Math. Hungar. 65 (2) (1994), 107-113. PRODUCT SETS IN THE PLANE SETS OF THE FORM A+B ON THE REAL LINE AND HAUSDORFF MEASURES Z. BUCZOLICH* (Budapest) Introduction. From Theorem 2 in [1] it follows that every E C [0, 1] x x [0, 1] = 12 of positive two-dimensional Lebesgue measure contains a set of the form A x B such that ,~I(A) > 0, and B is non-empty perfect. (We denote by ,~,~(A) the m-dimensional outer Lebesgue measure of the set A.) M. Laczkovich asked whether the set B can be of positive Hausdorff dimension. We show that the answer is negative. Moreover, in Theorem 1 we prove that for any Hausdorff measure nr there exists a set E C 12 of full measure such that if A x B C E, Aa(A ) > 0, and the sets A, B are measurable then B is of zero nr measure. (For the definition of the n r measure see the Preliminaries.) Sets of the form A+B = {a+b:a E A, b E B} can be regarded as projections of A x B onto the line y = x. G. Petruska asked the following question. Assume that ,~I(B) > 0 and the Hausdorff dimension of A n I equals d C [0, 1] for any interval [ r ~. Is it true that the gausdorff dimension of the complement of A + B cannot be bigger than 1 - d? In Theorem 2 we give a negative answer to this question. In fact we show that there exist B of full ,~a-measure, and a set A which satisfies the above conditions with d = 1 but the Hausdorff dimension of the complement of A + B also equals 1. Preliminaries. Assume that r [0,+ee) ~ [0,+ec) is monotone in- creasing, r > 0 for t > 0, r = 0, and r is continuous from the right for all t > 0. If E C [.Jl Ui and diam(Ui) __<5 (i = 1,2 .... ) then we say that the system {Ui} is a 5-cover of E. For an E C R put oo nO(E) = inf Z r Ui) i=1 where the inf is taken for all 5-covers of E. Put he(E) = sups>0 t~(E). It is well-known [2, Theorem 27, p. 50] that the Hausdorff measure t; r is a regular metric measure. Furthermore all Borel sets are he-measurable, and each he-measurable set of finite he-measure contains an Fo-set of the same measure. * Research supported by tile Hungarian National Foundation for Scientific Research Grant No. 2114. 0236-5294/94/$4.00 ~) 1994 Akad6iniai Kiad6, Budapest