One can have $|2A|$ as small as $2|A|$. Take $A = H \cup \{g\}$ where $H$ is a subgroup, $g \notin H$ and$g \neq -g$. Then $|A+A| = 2|A| + O(1)$ while $g+H$ and $H - g$ all have a unique ...

I am not sure of your level of knowledge of Sidon sets, but a good reference is O' Bryant's survey (see section 4.3 in particular) which contains several constructions in the integers. If you are ...

First off, we can remove the condition that $|A| \geq |F|^{\delta}$. One expects to be able to take any $\epsilon < 1$, as long as $|A| \leq |F|^{1/2}$ (not quite, see Oliver's comment below). The ...

His work with Dilworth, Ford, Konyagin, and Kutzarova solved an open problem in compressed image sensing. Thanks to this blog post of Tao from 2007, we can get a feel of the general mentality before ...

The critical value is $a= 5/2$. Chamizo and Cordoba showed that the fractal dimension of $f(x,a)$ is $2 + \frac{1}{2}(1/2 - a), \ \ \ 1 \leq a \leq \frac52$.Thus $f(x,a)$ is not differentiable for $a ...

Let $I(P,C)$ be the number of incidences in your point-circle configuration. One has $$I(P,C) \ll |P|^{2/3} |C|^{2/3} + |C| + |P|,$$ by the Szemeredi-Trotter theorem. Note that while this is optimal ...

We set $$e_N(\theta) := e^{2 \pi I \theta/N}.$$ Your sum which is equal to $$\left|\sum_x e_N(x^2 m)\right|^2$$ and is known as a Gauss sum. Gauss himself studied these extensively. Following him, we ...

The statement is not true. Take $B$ to be a set of the form $$\{a + b M : 1 \leq a, b \leq N\},$$ where $(N+1)M <p$, $N^2 \approx p^{1/3}$ and $n < 2M$. Then $B$ is Freiman isomorphic to the ...

For any fixed $k \geq 1$, one has $$D_k(n) \sim \frac{x}{\log x} \frac{(\log \log x)^{k-1}}{(k-1)!}.$$ This was originally proved by Landau in 1909 using induction of $k$. For $k \ll \log \log x$, ...

You should find this wiki page useful. The current unconditional record is 246. Assuming Elliot Halberstam, the current record is 12, and assuming generalized Elliot Halberstam, the current record is ...

As with most questions about finding primes in linear forms, nothing is known. We expect that the number of Sophie Germain primes $\leq x$ equivalent to 3 modulo 4 is $$\sim \frac{2x}{\log^2 x} \prod_{...

Actually, $|\sum_{M < n \leq N} e(x/n^2)| \sim c \sqrt{x}$, where $c \approx 0.016151690 + 0.0738060263i$. To see this, write $$\sum_{M < n \leq N} e(x/n^2) = \sum_{M < n \leq \epsilon \sqrt{...

I believe the eigenvectors are the ones you guessed, but in your second example, the dimensions of some of the eigenspaces are larger than one. I would guess that Mathematica chose a basis for those ...

First, tthere are other extremal examples than the complement of the Turan graph for $n = 3k-4$. The $3k-4$ example is given by the following. There is also an easy construction that shows that the ...

Seva's answer is in my opinion quite nice. I will elaborate on some other things that I have thought of to try to supplement what he has said. The Freiman dimension of an additive set, $A$, can be ...

This is a little late, but since this is the first thing that pops up with a google search of cliques in Paley graphs, I will mention the following: here is a result that improves the bound of $\sqrt{...

Here is a proof that $|X| \leq 4n-1$. Given your system $X$, let $\chi_1 , \ldots, \chi_N \in \mathbb{R}^{4n}$ be the characteristic vectors of the elements of $X$. Then \begin{equation*} \chi_i \...

I have always been impressed by the following construction of graphs with arbitrarily large chromatic number and girth. The construction I will mention is due to Nesetril-Rodl, and the the first such ...