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IV. Sequences without Arithmetic Progressions
48
Problems for Chapter IV
1. Attach a positive rational to each integer from 1 to 12 so that all
A.P.’s with common difference d up to 6 obtain their “correct”
measure 1 .
d
2. Prove that, if we ask for a generalization of this, then we can only
force the correct measure 1 for all A.P.’s of common difference
d √
d, by attaching weights onto 1, 2,...,è, if d O( n).
3. If we insist only on approximation, however, show that we can
always attach weights onto 1, 2,...,è such that the “measure”
given to every A.P. with common difference ≤ m is within e −n/m
of 1 .
a