No abstract

Let F 2 ZOEX; Y be an integral binary form of degree g > 2, and let ‰ F .x; y/ WD cardf1 6 a; b 6 x W P C .F .a; b// 6 yg where as usual P C .n/ denotes the largest prime factor of n. It is proved that ‰ F .x; y/ x 2 for y D x g 2C" in general, and y D x 1= p eC" if g D 3. Better results are obtained if F is reducible.

We define and study certain sum of digits functions in the context of finite fields. We give the number of polynomial values of F q with a fixed sum of digits. We also state a result for the sum of digits of polynomial values with generator arguments.

It is conjectured that the set G of the primitive roots modulo p has no decomposition (modulo p) of the form G = A+B with |A| > 2, |B| > 2. This conjecture seems to be beyond reach but it is shown that if such a decomposition of G exists at all, then |A|, |B| must be around p 1/2 , and then this result is applied to show that G has no decomposition of the form G = A + B + C with |A| > 2, |B| > 2, |C| > 2.

In an earlier paper Dartyge and Sárközy defined the measures of pseudorandomness of subsets of {1, 2, . . . , N}, and they presented several examples for subsets with strong pseudo-random properties. However, in applications one usually needs large families of subsets with strong pseudo-random properties. Here two constructions of this type are given. The notion of complexity of families of subsets of {1, . . . , N} is also introduced and studied.