Find a different parameter, show that it tends to infinity, and show that that implies that the discrepancy tends to infinity

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Introduction

This is a strategy that could potentially be applied either to the general problem or to the multiplicative case. The latter would probably be easier so I shall mainly discuss that case.

The thought behind it is that the discrepancy of a sequence is a pretty useless parameter in any kind of inductive argument. Note that in the multiplicative case, which is the one I am discussing at the moment, the discrepancy is simply the largest absolute value of any partial sum of the sequence. Now if you put together two sequences, what can you say? In order for this question to be even remotely sensible, when we talk about the discrepancy of the second sequence we must mean something like the largest drift along any segment of a HAP (where HAP is with respect to zero rather than to the point just before the second sequence starts). But knowing the discrepancy of the first sequence and the maximum drift of the second tells us virtually nothing about the discrepancy of the concatenation, since the HAPs where the maxima occur could have nothing to do with each other. And the same applies even if we put together several sequences: if we know that each sequence has a drift of 2C, say, there seems to be no way of using this information to prove that the concatenation of the sequences has drift greater than 2C.

To be continued.