Abstract An arc in $$\mathbb F_q^2$$ F q 2 is a set $$P \subset \mathbb F_q^2$$ P ⊂ F q 2 such that no three points of P are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let $${\mathcal {A}}(q)$$ A ( q ) denote the family of all arcs in $$\mathbb F_q^2$$ F q 2 . Our main result is the bound $$\begin{aligned} |{\mathcal {A}}(q)| \le 2^{(1+o(1))q}. \end{aligned}$$ | A ( q ) | ≤ 2 ( 1 + o ( 1 ) ) q . This matches, up to the factor hidden in the o(1) notation, the trivial lower bound that comes from considering all subsets of an arc of size q. We also give upper bounds for the number of arcs of a fixed (large) size. Let $$k \ge q^{2/3}(\log q)^3$$ k ≥ q 2 / 3 ( log q ) 3 , and let $${\mathcal {A}}(q,k)$$ A ( q , k ) denote the family of all arcs in $$\mathbb F_q^2$$ F q 2 with cardinality k. We prove that $$\begin{aligned} |{\mathcal {A}}(q,k)| \le \left( {\begin{array}{c}(1+o(1))q\\ k\end{array}}\right) . \end{aligned}$$ | A ( q , k ) | ≤ ( 1 + o ( 1 ) ) q k . This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound $$\begin{aligned} |{\mathcal {A}}(q,k)| \ge \left( {\begin{array}{c}q\\ k\end{array}}\right) \end{aligned}$$ | A ( q , k ) |

Let $A \subset \mathbb R$ and $G \subset A \times A$. We prove that for any $\lambda \in \mathbb R \setminus \{-1,0,1\}$, $ \max \{|A+_G A|, |A+_G \lambda A|, |A\cdot_G A|\} \gg |G|^{6/11}. $

It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$

This paper is a sequel to a paper entitled Variations on the sum-product problem by the same authors [SIAM J. Discrete Math., 29 (2015), pp. 514-540]. In this sequel, we quantitatively improve several of the main results of the first paper as well as generalize a method from it to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: $\exists a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}, |A(A-A)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{34}}, |A(A+A)| \gtrsim |A|^{\frac{3}{2}+\frac{5}{242}}, |\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$.

In this note it is established that, for any finite set A of real numbers, there exist two elements a, b from A such that |(a + A)(b + A)| > c|A|^2 / log |A|, where c is some positive constant. In particular, it follows that |(A + A)(A + A)| > c|A|^2 / log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev, which built upon the recent developments of Guth and Katz in their work on the Erdös distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemerédi-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from the paper of the author and Rudnev, since the set (a + A)(b + A) is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product A x A. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when A is a set of complex numbers.

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set $A\subset{\mathbb{R}}$, $|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$ giving a partial answer to a conjecture of Balog. In a similar spirit, it is established that $|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$ a bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing that $|A(A+a)|\gg{|A|^{3/2}}$ holds for a typical element of $A$.

An improved sum-product estimate for subsets of a finite field whose order is not prime is provided. It is shown, under certain conditions, that [Formula: see text]. This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev.