Digit Sum Bias and Nonvanishing Hankel Determinants

Definitions and mathematical statement

For an integer $n\geq 0$, write $n=\sum_{r\geq 0}\varepsilon_r2^r$, where $\varepsilon_r\in\{0,1\}$ and only finitely many digits are nonzero. Define its binary digit sum by $s_2(n)=\sum_{r\geq 0}\varepsilon_r$, so that $s_2(0)=0$. For integers $t\geq 1$ and $m\geq 0$, put \[ p_m(t)=2^{-m}\#\left\{n\in{\mathbb{Z}}:0\leq n<2^m,\ s_2(n+t)\geq s_2(n)\right\}. \] The source uses the limiting density \[ \gamma_\infty(t)=\lim_{m\to\infty}p_m(t). \] Existence of this limit for each positive integer $t$ is implicit in the source formulation; the supplied record does not provide a justification.

The digit-sum questions are:

1. Is it true that $\gamma_\infty(t)>\tfrac12$ for every integer $t\geq 1$?
2. What is the value of $\displaystyle\liminf_{\substack{t\to\infty\\t\in{\mathbb{Z}}_{\geq 1}}}\gamma_\infty(t)$?

For a sequence $a=(a_j)_{j\geq 0}$ of integers, define its shifted Hankel determinants by \[ H_{n,k}(a)=\det\left(a_{k+i+j}\right)_{0\leq i,j<n}, \qquad n\in{\mathbb{Z}}_{\geq 1},\quad k\in{\mathbb{Z}}_{\geq 0}. \] The further existence question is whether there are a finite nonempty set $A\subseteq{\mathbb{Z}}$ and an infinite sequence $a\in A^{{\mathbb{Z}}_{\geq 0}}$ such that \[ \forall n\in{\mathbb{Z}}_{\geq 1}\ \forall k\in{\mathbb{Z}}_{\geq 0},\qquad H_{n,k}(a)\neq 0. \]

The supplied registry labels the combined record partially solved. Its dated assessment attributes a proof of the universal strict digit-sum density inequality to Cheng and describes the requested limit inferior and the Hankel-determinant existence question as still open. These are supplied literature-status claims, not independently verified mathematical conclusions of this draft.

Short English statement

For each positive shift, compare the binary digit sums of an integer and its translate using densities on intervals $[0,2^m)$. Ask whether every limiting density exceeds $1/2$, and determine their limit inferior as the shift tends to infinity through positive integers. Also ask whether an infinite sequence taking only finitely many integer values can have every shifted Hankel determinant nonzero. The supplied dated assessment labels the first question solved and the other two open; this draft does not verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001559$, title Digit Sum Bias and Nonvanishing Hankel Determinants, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Combinatorics on Words (2011), Oberwolfach Reports, pp. 2195--2244, DOI 10.4171/owr/2010/37. https://doi.org/10.4171/owr/2010/37. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. It separately gives proposed year 2010; the relationship between these year labels is unresolved here.
• Supplied literature reference: Cheng, 2026. https://arxiv.org/abs/2606.23398. The supplied assessment attributes a proof of the universal Cusick density inequality to this reference; that attribution has not been independently verified.
• Supplied literature reference, labelled "current open-problems survey": https://cs.uwaterloo.ca/~shallit/Talks/bc4.pdf. No further bibliographic identification is supplied.
• Supplied literature reference, labelled "original workshop report": https://doi.org/10.4171/owr/2010/37.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the record partially solved, states that Cheng proves the universal Cusick density inequality, and states that the requested limit inferior and finite-alphabet sequence with every Hankel determinant nonzero remain open. The supplied background explicitly calls this dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.