On the slim exceptional set for the Lagrange four squares theorem

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E 3(N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P 5. Then we have E 3(N)≪log1053 N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P 11 in place of P 5.