Determining $H_0$ from distance sum rule combining gamma-ray bursts with observational Hubble data and strong gravitational lensing

Model-independent bounds on the Hubble constant $H_0$ are important to shed light on cosmological tensions. We work out a model-independent analysis based on the sum rule, which is applied to late- and early-time data catalogs to determine $H_0$. Through the model-independent B\'ezier interpolation of the observational Hubble data (OHD) and assuming a flat universe, we reconstruct the dimensionless distances of the sum rule and apply them to strong lensing data to derive constraints on $H_0$. Next, we extend this method to the high-redshift domain including, in other two separated analyses, gamma-ray burst (GRB) data sets from the well-established Amati and Combo correlations. In all three analyses, our findings agree at $1\sigma$ level with the $H_0$ determined from type Ia supernovae (SNe Ia), and only at $2\sigma$ level with the measurement derived from the cosmic microwave background (CMB) radiation. Our method evidences that the bounds on $H_0$ are significantly affected by strong lensing data, which favor the local measurement from SNe Ia. Including GRBs causes only a negligible decrease in the value of $H_0$. This may indicate that GRBs can be used to trace the expansion history and, in conjunction with CMB measurements, may heal the Hubble tension and accommodate to the flat $\Lambda$CDM paradigm purported by CMB data.