In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.

Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$\left|C\right|>\left\{\begin{array}{cc}\lfloor n/2\rfloor +1\hfill & \text{if}m\text{is}\text{odd},\hfill \\ m\ell /2+\delta \hfill & \text{if}m\text{is}\text{even},\hfill \end{array}\right.$$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\delta$ denotes the cardinality of even numbers in the interval $[0,min\{r,m-2\left\}\right]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given $c>0$, there is...

We describe all sets $A{\subseteq }_p$ which represent the quadratic residues $R{\subseteq }_p$ in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.