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arXiv · 2606.24195

Tractability versus curse of dimensionality for geometric $L_p$-discrepancies

Erich Novak·Friedrich Pillichshammer
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Abstract

This paper studies tractability versus the curse of dimensionality for several geometric $L_p$-discrepancies through a unified discrepancy--integration duality framework, where worst case integration errors in suitable function spaces equal the corresponding discrepancies. A general lower bound method for non-negative linear rules in spaces under broad tensor-product assumptions establishes exponential information complexity in the dimension $d$, yielding the curse of dimensionality for the respective discrepancy. We complement this overview by new results on discrepancy--integration duality and the curse of dimensionality for the periodic $L_p$-discrepancy. The current state of research on this general problem is summarized in a clearly laid out table, which also highlights the remaining open questions.

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Erich Novak, Friedrich Pillichshammer. 2026-06-23. Tractability versus curse of dimensionality for geometric $L_p$-discrepancies. https://arxiv.org/abs/2606.24195

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