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Given a smooth integer q, we use existing upper bounds for character sums to find a lower bound for the size of a multiplicative subgroup of the integers modulo q which contains the image of an interval of consecutive integers I ⊂ Zq under a polynomial f ∊ Z[X]. [ABSTRACT FROM AUTHOR]

We prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P1, . . ., P4 ∈ C², and Q1, . . .,Qn ∈ C², if there are C² n(1+δ)/2 distinct lines between Pi and Qj for all i, j, then P1, . . ., P4 are collinear. If the number of the distinct lines is < cn1/2, then the cross ratio of the four points is algebraic. 2. Given c > 0, there is δ > 0 such that for any P1, P2, P3 ∈ C² noncollinear, and Q1, . . .,Qn ∈ C², if there are ≤ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P ∈ C² r {P1, P2, P3}, we have δn distinct lines between P and Qj. 3. Given c > 0, there is ∊ > 0 such that for any P1, P2, P3 ∈ C² (respectively, R²) collinear, and Q1, . . .,Qn ∈ C² (respectively, R²), if there are ∊ cn1/2 distinct lines between Pi and Qj for all i, j, then for any P not lying on the line L(P1, P2), we have at least n1-∊ (resp. n/log n) distinct lines between P and Qj. The main ingredients used are the subspace theorem, Balog-Szemerédi-Gowers theorem, and Szemer 'edi-Trotter theorem. We also generalize the theorems to higher dimensions, extend Theorem 1 to F²p, and give the version of Theorem 2 over Q. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate efficient representations of the residue classes modulo q, by performing sum and product set operations starting from a given subset A of ℤq. We consider the case of very small sets A and composite q for which not much seemed known (non-trivial results were recently obtained when q is prime or when log ∣A∣ ∼ log q). Roughly speaking we show that all residue classes are obtained from a k-fold sum of an r-fold product set of A, where r ⟪ log q and log k ⟪ log q, provided the residue sets πq′,(A) are large for all large divisors q′ of q. Even in the special case of prime modulus q, some results are new, when considering large but bounded sets A. It follows for instance from our estimates that one can obtain r as small as r ∼ log q/log ∣A∣ with similar restriction on k, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking for A = H a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on mina∊ℤq* maxx∊H ∥ax/q∥ are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of xm = a (mod q) to composite modulus q. [ABSTRACT FROM AUTHOR]