Math. Proc. Camb. Phil. Soc. (1994), 115, 437 437 Printed in Great Britain The packing measure of self-affine carpets BY YUVAL PERES Yale University, Department of Mathematics, New Haven, CT 06520 {Received 5 May 1993; revised 26 August 1993) Abstract We show that the self-affine sets considered by McMullen[15] and Bedford [2] have infinite packing measure in their packing dimension 8 except when all non-empty rows of the initial pattern have the same number of rectangles. More precisely, the packing measure is infinite in the gauge ^log^l" 1 and zero in the gauge ^log^l" 1 '* for any S > 0. 1. Introduction Besides the notion of Hausdorff dimension dim H , another frequently used notion of dimension is the Minkowski (or 'box') dimension: For a totally bounded set E in a metric space, its (upper) Minkowski dimension is L , (1-1) 40 ~~ lu & fc where N(E, e) denotes the largest possible number of disjoint balls of diameter e centred at points of E. However, this notion suffers from the lack of associated measures. Tricot[21, 22] introduced packing dimension, which is dual to Hausdorff dimension in several senses and does come with associated measures. For any increasing funtion <f>: [0, oo) -> R such that 0(0) = 0 and any set E ina, metric space, define first the packing premeasure (in gauge <j>) by f 1 ^ = lim sup £ ^(diam^.) , (1-2) where the supremum is over all collections of disjoint closed balls {Bj}f =l with centres in E and diameters diam (Bj) < e. Then define the packing measure in gauge (/> by % t i ) (1-3) 4-1 t=l J This is indeed a Borel measure; when (p(t) = f we write 0> d for 0^ {$P g is called 6- dimensional packing measure). Finally, define the packing dimension of E by dim P (#) = inf{0: %(E) = 0}. (1-4) We always have d\m H {E) ^ Aim P (E) ^ dIm M (S); other general properties of packing measures are recalled in Section 2.