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The Erdős discrepancy problem asks, “Does there exist an infinite sequence (ti)i⩾1taking values in {−1, 1} and a constant csuch that |∑1⩽i⩽ntid|⩽cfor all ?” Erdős conjectured in the 1930s that no such sequence exists. We examine some variations of this problem with fixed values for cin which the values of dare restricted to particular subsets of . When the values of dare restricted to powers of 2, we show that there are exactly two infinite sequences with discrepancy bounded by 1 and an uncountable number of infinite sequences with discrepancy bounded by 2. When the values of dare restricted to the powers of bfor b>2, we show that there is an uncountable number of infinite sequences with discrepancy bounded by 1. We also give a recurrence for the number of sequences of length nwith discrepancy bounded by 1. When the values of dare restricted to the odd numbers, we conjecture that there are exactly four infinite sequences with discrepancy bounded by 1 and give some experimental evidence for this conjecture.