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- W2004251073 abstract "Sequences of integers exist with any possible upper density and lower density which do not contain an infinitely long arithmetic progression. A well-known conjecture in additive number theory states that for every integer k greater than 1, if an increasing sequence of positive integers does not contain k consecutive terms of an arithmetic progression, then the sequence has density zero. Much work has been done on this problem during the last 40 years, but we have never seen it remarked that the analogous conjecture for infinite arithmetic progressions is false. Namely it is not true that if an increasing sequence of positive integers does not contain an infinitely long arithmetic progression, then the sequence has density zero. (The converse is obviously true.) The following easy theorem shows that the statement is very far from true. Theorem. Let a and b be real numbers such that 05=a^ei£l. Then there exists an increasing sequence A of positive integers which has lower density a and upper density b and which contains no infinitely long arithmetic progression. Proof. The sequence A will contain long blocks of consecutive integers separated by large gaps. It is defined by *, + 1, kx + 2, • • • , kx + lx, kx + lx + k2 + 1, • • , kx + lx + k2 + l2, kx + lx + k2 + l2 + k3 + , ■ ■ ■, where the positive integers kx, lx, k2, l2, • ■ • will be defined later. Let {an} and (bf be sequences of real numbers satisfying linv^ an=a, Iim,,^^ bn=b and (1) 0 n-i'2 Received by the editors February 29, 1972. AMS (MOS) subject classifications (1970). Primary 10L10." @default.
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- W2004251073 title "Sequences not containing an infinite arithmetic progression" @default.
- W2004251073 doi "https://doi.org/10.1090/s0002-9939-1972-0308074-6" @default.
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