For any two \(t\)-structures \(\mathcal U\) and \(\mathcal V\) on a triangulated category \( \mathcal D\), the distance \(d(\mathcal U,\mathcal V)\) is the smallest natural integer \(d\) such \(\mathcal V_{\leq m}\subseteq \mathcal U_{\leq 0}\subseteq \mathcal V_{\leq m+d}\) for some integer \(m\). If there is no such number \(d\), then the distance \(d(\mathcal U,\mathcal V)\) is infinity [Y. Qiu and J. Woolf, Geom. Topol. 22, No. 6, 3701–3760 (2018; Zbl 1423.18044)]. Given a triangulated category \(\mathcal T\) with a \(t\)-structure \(\mathcal U\), let \(\mathcal T^b\) be the smallest triangulated full subcategory containing the heart of \(U\). By [A. A. Beilinson et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)], \(\mathcal U\) restricts to a bounded t-structure on \(\mathcal T^b\), denoted by \(\mathcal U^b\).
In [B. Keller, Doc. Math. 10, 551–581 (2005; Zbl 1086.18006)], there is statement without proof: For any \(t\)-structure \(\mathcal V'\) on \(\mathcal T^b\) satisfying \(d(U^b,\mathcal V')\) extends canonically to a \(t\)-structure \(\mathcal V\) on \(\mathcal T\). Base on the analysis of global dimensions and distance of t-structures \(\mathcal U\), the authors give a stronger formulation by establishing a bijective correspondence between the set of \(t\)-structures \(\mathcal V\) on \(\mathcal T\) with \(d(\mathcal U,\mathcal V)<+\infty\) and the set of t-structures \(\mathcal V'\) on \(\mathcal T^b\) with \(d(\mathcal U^b,\mathcal V')<+\infty\).