Irregularities of sequences relative to arithmetic progressions, III

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Abstract

It is conjectured that an integer sequence containing no k consecutive terms of any arithmetic progression must have density zero. Only the cases k = 3 [Roth (1952) by an analytic method] and k = 4 [Szemerédi (1967) by an elementary method] have so far been settled. The basis of Roth's method was a result concerning the exponential sums associated to finite sequences having the property in question. In the present paper we establish a corresponding (but different and more elaborate) result which will enable us to incorporate Szemerédi's ideas in an analytic method.

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University of Colorado, Boulder, February–June, 1969.