Large family of pseudorandom subsets of the set of the integers not exceeding \(N\)
The author studies the pseudorandomness of the sets NEWLINE\[NEWLINE{\mathcal R}=\{1\leq n<p : \exists r\leq h<r+s:f(n)+n^{-1}\equiv h\mod p\},NEWLINE\]NEWLINE where \(p\) is a prime, \(f\) a polynomial over the finite field of \(p\) elements, and \(r,s\) are integers with \(1\leq s<p\). Using some standard character sum bounds, he estimates two measures of pseudorandomness introduced by \textit{C. Dartyge} and \textit{A. Sárközy} [Period. Math. Hung. 54, No. 2, 183--200 (2007; Zbl 1174.05001)], the well-distribution measure and the correlation measure of order \(k\) of \({\mathcal R}\). Moreover, he also studies the analogous question for composite modulus \(pq\) with two distinct odd primes and provides bounds on the well-distribution measure and correlation measure of order \(2\) and \(3\). (Note that there is no non-trivial bound on the correlation measure of order \(4\) taking lags \(0\), \(p\), \(q\), and \(p+q\).)
scientific article; zbMATH DE number 6331996