Bounds on 10th moments of (x, x^3) for ellipsephic sets

Abstract

Let A be an ellipsephic set which satis es digital restrictions in a given base. Using the method developed by Hughes and Wooley, we bound the number of integer solutions to the system of equations X2 i=1 ð€€€ x3i ð€€€ y3 i

= X5 i=3 ð€€€ x3i ð€€€ y3 i

X2 i=1 (xi ð€€€ yi) = X5 i=3 (xi ð€€€ yi); with x; y 2 A5. The fact that ellipsephic sets with small digit sumsets have fewer solutions of linear equations allows us to improve the general bounds obtained by Hughes andWooley and also the corresponding e cient congruencing estimates. We also generalize our result from the curve (x; x3) to (x; (x)), where is a polynomial with integer coe cients and deg( ) 3.