Asymptotically automatic sequences generalize the notion of automatic sequences by relaxing the requirement of finite k-kernels to finite up to asymptotic equality. This paper systematically investigates the properties of asymptotically automatic sequences and establishes analogues or counterparts to various well-known facts about automatic sequences.
Weil sums of binomials with four distinct values must be rational integers, except in the case where the finite field has order 5 and the exponent is congruent to 3 modulo 4.