On an application of higher energies to Sidon sets

We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum--products results, improving a result of Balog and Wooley, in particular.

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.

Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.

We say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\{1,2,\dots,n\}$ is at most $\pi(n)+\pi(n/2)+n^{2/3}(\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $\pi(n)+\pi(n/2)+cn^{2/3}\log n/\log\log n$.

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the lower order term: $\pi(n)+\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\{1,\ldots, n\}$ is $T(n)\cdot 2^{\Theta(\frac{n^{3/4}}{(\log n)^{3/2}})}$ for a certain function $T(n)\approx 2^{1...

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.

We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tup...

A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach, we prove that there exists a perfect difference set A such that A(x) >> x^{\sqrt{2}-1-o(1)}. We also prove that there exists a perfect difference set A such that limsup_{x\to \infty}A(x)/\sqrt x\geq 1/\sqrt 2.

Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called Sidon set if all the two terms sums formed by the elements of $A$ are different. Many years ago P. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os asked whether there exists a Sidon set which is asymptotic basis of order $3$. In this paper we prove t...

A set $S\subset\{1,2,...,n\}$ is called a Sidon set if all the sums $a+b~~(a,b\in S)$ are different. Let $S_n$ be the largest cardinality of the Sidon sets in $\{1,2,...,n\}$. In a former article, the author proved the following asymptotic formula $$\sum_{a\in S,~|S|=S_n}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$$ where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an extension of the above formula. We show that $$\sum_{a\in S,~|S|=S_n}a^{\ell}=\...