Redundancy of the integer-intersection hypothesis

Prove that the assumption $T(b,-)\cap\mathbb Z=\{0\}$ in part (3) of Proposition 2.12 is redundant.

Background

Proposition 2.12 identifies the smallest period of an integer cycle associated with a primitively incomplete integer, under the additional assumption that the auxiliary self-similar set T(b,−)T(b,-) has no nonzero integer points.

The authors explicitly conjecture that this auxiliary assumption is unnecessary, so that the stated period conclusion should hold without it.

References

We guess that the assumption $T(b, -)\cap ={0}$ in (3) of Proposition \ref{prop2.12} is redundant.

— Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture  (2609.29038 - He et al., 24 Sep 2026) in Remark following Proposition 2.12, Section 2