Minimum lengths of sign sequences with prescribed fairness exponent

Determine the minimum lengths of sign sequences whose fairness exponent is at least j for all j beyond the currently computed range, equivalently the minimum degrees of Littlewood polynomials having a root of multiplicity at least j at 1.

Background

The paper associates a finite sign sequence with a generating polynomial and defines its fairness exponent as the multiplicity of the root at 1 of the corresponding fairness polynomial. The Thue–Morse block T_j has fairness exponent j, but it is not always a shortest sign sequence with that property: a length-48 example has multiplicity 6, whereas T_6 has length 64.

The minimum lengths are known for fairness exponents j=1 through 10, with values 2, 4, 8, 16, 32, 48, 96, 144, 192, and 240. The authors state that the values for larger j remain unknown. This is equivalently a problem about determining the minimum degree of a Littlewood polynomial with a root of multiplicity at least j at 1.

References

The minimum lengths for $j=1,\ldots,10$ are $2, 4, 8, 16, 32, 48, 96, 144, 192, 240$; for larger $j$, they remain unknown (see and the references therein).

— Fair Duels after a False Start  (2610.00880 - Khovanova et al., 1 Oct 2026) in Section 2, immediately after Lemma 2.2

This leads to a natural open question: what is the asymptotic growth of $\lambda(i)$? Our counting heuristic suggests $\lambda(i)=\Theta(i3)$, corresponding to a maximum least period of $2{\Theta(n{1/3})}$, but the influence of the arithmetic restrictions on this prediction remains unclear.

— Fair Duels after a False Start  (2610.00880 - Khovanova et al., 1 Oct 2026) in Section 7, Final Remarks