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<p>In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.</p>

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On a Discrete Approach to Lower Bounds in Discrepancy Theory

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月11日

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  • Luca Brandolini,
  • Bianca Gariboldi,
  • Giacomo Gigante,
  • Alessandro Monguzzi

摘要

In this paper, we prove that some renowned lower bounds in discrepancy theory admit a discrete analogue. Namely, we prove that the lower bound of the discrepancy for corners in the unit cube due to Roth holds true also for a suitable finite family of corners. We also prove two analogous results for the discrepancy on the torus with respect to squares and balls.