Underlying pattern in the locations of extreme-value clusters

Identify a general rule determining the locations of the narrow subintervals in which the extreme values of the Gauss circle error term \(E(t)\) are concentrated.

Background

The computations show that the extreme values of E(t)E(t), represented by the record functions emax(t)emax(t) and einf(t)einf(t), are not uniformly distributed across the computed range. Instead, they cluster in several narrow subintervals. The paper records this phenomenon and gives computationally observed localization intervals, but explicitly states that no general rule for locating those intervals is known.

References

Although these subintervals were identified through numerical experimentation, we do not know of any general rule for determining their locations.

— Numerical experiments on the Hardy conjecture for the Gauss circle problem  (2609.04725 - Yamaguchi et al., 4 Sep 2026) in Section 1, Introduction

However, we were unable to identify any underlying pattern in the sequences $S_{\max}$ and $S_{\inf}$.

— Numerical experiments on the Hardy conjecture for the Gauss circle problem  (2609.04725 - Yamaguchi et al., 4 Sep 2026) in Section 5, "The tendency of the points in $T_{\pm}$ to cluster", Remark following Proposition 5.1