Elsevier

Theoretical Computer Science

Volume 123, Issue 1, 17 January 1994, Pages 139-144
Theoretical Computer Science

Multiple returns under a bounded number of iterations

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Abstract

Kowalski (1991) obtained the quantity c(k, r, a), which satisfies the condition max {m(Afi1 A∩ ⋯ ∩ f−ir A): 1 ⩽ i1 < ⋯ < irk} ⩾ c(k, r, a) for any dynamical system (X,B, m, f) and every set AB, with m(A) = a.

Our present purpose is to prove the following theorem: for every ergodic aperiodic endomorphism and for every integer k, and for each real a, 0 < a < 1, such that (k + 1)a is not an integer, there exists a set A with m(A) = a fulfilling the equality m(Afi1 A∩⋯∩fir A) = c(k, r, a), for any sequence 1 ⩽ i1 < ⋯ < irk, r = 1, …, k.

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Supported by KBN Grant PB-666/2/91.