PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 3, March 1998, Pages 739–744 S 0002-9939(98)04082-9 SMORODINSKY’S CONJECTURE ON RANK-ONE MIXING TERRENCE M. ADAMS (Communicated by Mary Rees) Abstract. We prove Smorodinsky’s conjecture: the rank-one transformation, obtained by adding staircases whose heights increase consecutively by one, is mixing. 0. Introduction The first rank-one mixing transformation was constructed by Ornstein [O] using “random” spacers on each column. We refer to [F1] for a description of rank-one constructions. Recently, the first rank-one mixing transformation was constructed with an explicit formula for adding spacers [AF]. In [AF] a method for adding staircases was given which produced mixing. However, Smorodinsky’s conjecture re- mained open. M. Smorodinsky conjectured that by adding staircases whose heights increase consecutively by one, the resulting transformation (classical staircase con- struction) is mixing. In this paper, we will prove that an infinite staircase construction, whose se- quence r n of cuts and h n of heights satisfy the condition lim n→∞ r 2 n hn = 0, is mixing. Thus Smorodinsky’s conjecture follows as a corollary. 1. Staircase constructions A rank-one transformation T is called a staircase construction if there exists a sequence (r n ) ∞ n=1 of natural numbers such that each column C n+1 is obtained by cutting C n into r n subcolumns of equal width, placing i − 1 spacers on the i th subcolumn for 1 ≤ i ≤ r n , and then stacking the (i + 1) st subcolumn on top of the i th subcolumn for 1 ≤ i ≤ r n . Denote T = T (rn) . Let h n be the height of column C n for n ≥ 1. From now on, we assume the sequence (h n ) is derived from the sequence (r n ) in this manner. If the sequence r n is bounded then T is called a finite staircase construction. The classical staircase construction is given by r n = n. If r n →∞, we call T an infinite staircase construction. In this case T may not be finite measure preserving since we may be adding measure too quickly. Assume T is finite measure preserving. The following question remains open: Question. Is every infinite staircase construction mixing? Received by the editors August 20, 1996. 1991 Mathematics Subject Classification. Primary 28D05. Key words and phrases. Mixing, rank-1. c 1998 American Mathematical Society 739 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use