Chebyshev's bias for products of irreducible polynomials

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.Comment: The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bia

Asymptotic estimate on the distance energy of lattices

Since the well-known breakthrough of L. Guth and N. Katz on the Erdos distinct distances problem in the plane, mainstream of interest is aroused by their method and the Elekes-Sharir framework. In short words, they study the second moment in the framework. One may wonder if higher moments would be more efficient. In this paper, we show that any higher moment fails the expectation. In addition, we show that the second moment gives optimal estimate in higher dimensions.Comment: We show that the higher moments in Guth-Katz framework of Erdos distinct distances do not help, and some mor