INTERSECTING RATIONAL BEATTY SEQUENCES Jamie
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چکیده
A rational Beatty sequence has the form {�pi/q + b� : i ∈ Z} where p > q > 0 and gcd(p, q) = 1. We call p/q the modulus of the sequence and b the offset. Morikawa gave a condition on the moduli of two Beatty sequences such that they would be disjoint for a suitable choice of offsets. Holzman and Fraenkel showed that the sequence formed by the intersection of two Beatty sequences with moduli p1/q1 and p2/q2, q2 ≤ q1, could have as many as q2 +3 distinct consecutive differences. In this note we show that if the moduli satisfy the Morikawa condition but the sequences do intersect then the consecutive differences take on at most three different values.
منابع مشابه
A 50 Integers 13 ( 2013 ) Intersecting Rational Beatty Sequences
A rational Beatty sequence has the form {�pi/q + b� : i ∈ Z} where p > q > 0 and gcd(p, q) = 1. We call p/q the modulus of the sequence and b the offset. Morikawa gave a condition on the moduli of two Beatty sequences such that they would be disjoint for a suitable choice of offsets. Holzman and Fraenkel showed that the sequence formed by the intersection of two Beatty sequences with moduli p1/...
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تاریخ انتشار 2013