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We study the distribution of   S  ‐integral points on    SL  2   ‐orbit closures of binary forms and prove an asymptotic formula for the number of   S  ‐integral points. This extends a result of Duke, Rudnick and Sarnak. The main ingredients of the proof are the method of mixing developed by Eskin–McMullen and Benoist–Oh, Chambert‐Loir–Tschinkel's study of asymptotic volume of height balls and Hassett–Tschinkel's description of log resolutions of    SL  2   ‐orbit closures of binary forms.

The distribution of S ‐integral points on SL 2 ‐orbit closures of binary forms