We construct smooth projective varieties of general type with the smallest known volumes in high dimensions. Among other examples, we construct varieties of general type with many vanishing plurigenera, more than any polynomial function of the dimension. As part of the construction, we solve exactly an optimization problem about equidistribution on the unit circle in terms of the sawtooth (or signed fractional part) function. We also solve exactly the analogous optimization problem for the sine function. Equivalently, we determine the optimal inequality of the form $sum_{k=1}^m a_ksin kxleq 1$, in the sense that $sum_{k=1}^m a_k$ is maximal.
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