This was asked by Erdős in 1986 at Oberwolfach.

Erdős proved that\[\delta_1(n,m) \ll \frac{1}{(\log n)^c}\]for all $m$, for some constant $c>0$. Sharper bounds (for various ranges of $n$ and $m$) were given by Ford [Fo08].

Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$. For example,\[\delta_1(3,6)= 0.35\quad \delta_1(3,7)\approx 0.33\quad \delta_1(3,8)\approx 0.3619.\]Furthermore, Cambie [Ca25] has shown that, for fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$.

See also [446].

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This page was last edited 04 November 2025.