Related papers: On sumsets containing a perfect square

It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={\pi^2/6}$. Meir and Moser asked what is the smallest $\epsilon$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ can be packed into a rectangle of area…

Euler explored the problem of finding three numbers such that the sum or difference of any two of them is a perfect square. He discovered a parametric solution represented by polynomials of degree 18 and identified the smallest of these…

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le…

Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of…

We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.

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

We classify the sum-free subsets of ${\mathbb F}_3^n$ whose density exceeds $\frac16$. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset ${A\subseteq {\mathbb F}_3^n}$ is maximal with…

Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t <…

For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically,…

Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented…

A perfect matching in a 3-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a 3-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} -…

Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha)$ in the inequality…

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if and only if certain nonnegative…

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $. This result essentially is proved -- with the same tools -- by Mei-Chu Chang. See in J. Funct. Anal. 207 (2004), no 2,…

Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(\alpha;N)=\sum_{b\in B,b\leq N} e(b\alpha)$ satisfies \[ \frac{S(\alpha; N)}{N/\sqrt{\log N}}<<_{A,A'} \frac{1}{\phi(q)}…

We obtain an upper bound for the number of pairs $ (a,b) \in {A\times B} $ such that $ a+b $ is a prime number, where $ A, B \subseteq \{1,...,N \}$ with $|A||B| \, \gg \frac{N^2}{(\log {N})^2}$, $\, N \geq 1$ an integer. This improves on a…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are…

Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares…

Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the…