The Falconer distance problem for Cartesian product sets was introduced and studied by Iosevich and Liu ([13]). In this paper, by implementing a new observation on Cartesian product sets associated with a particular parabolic structure, we study the pinned version of Falconer distance problem for Cartesian product sets, and improve the threshold for the Falconer distance set in [13] in certain cases.

The authors are grateful to the anonymous referee for valuable comments and helpful suggestions. J. Li is supported by ARC DP 260100485. C.-W. Liang and C.-Y. Shen are supported in part by NSTC through grant 111-2115-M-002-010-MY5. C.-W. Liang is also supported by MQ Cotutelle PhD scholarhsip.