Sum-free Sets Research Articles

The Structure of Large SUM-Free Sets in 𝔽n p

ABSTRACT A set $A\subset \mathbb{F}_p^n$ is sum-free if A + A does not intersect A. If $p\equiv 2 \;\mathrm{mod}\; 3$, the maximal size of a sum-free set in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset \mathbb{F}_p^n$ has size at least $p^n/3-p^{n-1}/6+p^{n-2}$, then there exists subspace $V\lt\mathbb{F}_p^n$ of codimension 1 such that A is contained in $(p+1)/3$ cosets of V. For p = 5 specifically, we show the stronger result that every sum-free set of size larger than $1.2\cdot 5^{n-1}$ has this property, thus improving on a recent theorem of Lev.

Sidon sets, sum-free sets and linear codes

Sidon sets, sum-free sets and linear codes

Research on the sumsets of different types of group

In this article, author introduce the theories about the sums of sets. First the definitionX Y = { x_i y_j | x_i X,y_jY }, that X, Y is a subset of group G. the first relative theorem is prove by CauchyDavenport. The beginning is the Knesers theory, and then, in the second part, this paper introduces several types of sumsets, though most of them is based on abelian group (commutative group or additive group) and if not, the author gives out the definition. And the popular topic about this is the sum-free sets, but this paper will only introduce a few. Readers who want detailed prove need to look for proves on the reference articles and the theory author talked is a small part of them (important or interesting ones). In some of references, the + can be change to other operation which satisfy the group, so this paper changes them into .

Integer colorings with forbidden rainbow sums

For a set of positive integers A⊆[n], an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of [n] with the maximum number of rainbow sum-free r-colorings. We show that for r=3, the interval [n] is optimal, while for r≥8, the set [⌊n/2⌋,n] is optimal. We also prove a stability theorem for r≥4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.</p>

Tensor slice rank and Cayley's first hyperdeterminant

Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes.

Shape of the asymptotic maximum sum-free sets in integer lattice grids

We determine the shape of all sum-free sets in {1,2,…,n}2 of size close to the maximum 35n2, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe 45n−o(n)≤x+y≤85n+o(n). We also determine for any positive integer p the maximum size of a subset A⊆{1,2,…,n}2 which forbids the triple (x,y,z) satisfying px+py=z.

On Maximal Sum-Free Sets in Abelian Groups

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erdős. In contrast, not as much is know about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown concerning a general lower bound on the number of maximal sum-free sets in abelian groups of a fixed order. Indeed, we verify the conjecture for all finite abelian groups with a cyclic component of size at least 3084. Other related results and open problems are also presented.

Regular saturated graphs and sum-free sets

In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular F-saturated graphs. One of the essential questions is given F, for which n does a regular n-vertex F-saturated graph exist. They proved that for all sufficiently large n, there is a regular K3-saturated graph with n vertices. We extend this result to both K4 and K5 and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all k≥2, there is a regular C2k+1-saturated with n vertices for infinitely many n. Studying the sum-free sets that give rise to C2k+1-saturated graphs is an interesting problem on its own and we state an open problem in this direction.

The largest $(k,\ell )$-sum-free subsets

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell)$.

On sets with small sumset and m-sum-free sets in Z/pZ

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is covered by an arithmetic progression of size at most $|A|+r+1$. A theorem of Serra and Zemor proves the conjecture provided $r\leq 0.0001|A|$, without any additional constraint on $|A|$. Subject to the mild additional constraint $|2A|\leq 3p/4$ (which is optimal in a sense explained in the paper), our first main result improves the bound on $r$, allowing $r\leq 0.1368|A|$. We also prove a variant which further improves this bound on $r$ provided $A$ is sufficiently dense. We then give several applications. First we apply the above variant to give a new upper bound for the maximal density of $m$-sum-free sets in $\mathbb{Z}/p\mathbb{Z}$, i.e., sets $A$ having no solution $(x,y,z)\in A^3$ to the equation $x+y=mz$, where $m\geq 3$ is a fixed integer. The previous best upper bound for this maximal density was $1/3.0001$ (using the Serra-Zemor Theorem). We improve this to $1/3.1955$. We also present a construction following an idea of Schoen, which yields a lower bound for this maximal density of the form $1/8+o(1)_{p\to\infty}$. Another application of our main results concerns sets of the form $\frac{A+A}{A}$ in $\mathbb{F}_p$, and we also improve the structural description of large sum-free sets in $\mathbb{Z}/p\mathbb{Z}$.

On those multiplicative subgroups of $${\mathbb F}_{2^n}^*$$ which are Sidon sets and/or sum-free sets

We study those multiplicative subgroups of $${\mathbb F}_{2^n}^*$$ which are Sidon sets and/or sum-free sets in the group $$({\mathbb F}_{2^n},+)$$ . These Sidon and sum-free sets play an important role relative to the exponents of APN power functions, as shown by a paper co-authored by the first author.

Groups with few maximal sum-free sets

A set of integers is sum-free if it does not contain any solution for x+y=z. Answering a question of Cameron and Erdős, Balogh, Liu, Sharifzadeh and Treglown recently proved that the number of maximal sum-free sets in {1,…,n} is Θ(2μ(n)/2), where μ(n) is the size of a largest sum-free set in {1,…,n}. They conjectured that, in contrast to the integer setting, there are abelian groups G having exponentially fewer maximal sum-free sets than 2μ(G)/2, where μ(G) denotes the size of a largest sum-free set in G.We settle this conjecture affirmatively. In particular, we show that there exists an absolute constant c>0 such that almost all even order abelian groups G have at most 2(1/2−c)μ(G) maximal sum-free sets.

L-functions and sum-free sets

For set \(A\subset \mathbb{F}_p^*\) define by \(\mathsf{sf}(A)\) the size of the largest sum-free subset of A. Alon and Kleitman [3] showed that \(\mathsf{sf} (A) \ge |A|/3+O(|A|/p)\). We prove that if \(\mathsf{sf}(A)-|A|/3\) is small then the set A must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in \(\mathbb{F}p\).

Sum-free sets generated by the period-[formula omitted]-folding sequences and some Sturmian sequences

First, we show that the sum-free set generated by the period-doubling sequence is not κ-regular for any κ≥2. Next, we introduce a generalization of the period-doubling sequence, which we call the period-k-folding sequences. We show that the sum-free sets generated by the period-k-folding sequences also fail to be κ-regular for all κ≥2. Finally, we study the sum-free sets generated by Sturmian sequences that begin with ‘11’, and their difference sequences.

A DISTRIBUTION ON TRIPLES WITH MAXIMUM ENTROPY MARGINAL

We construct an $S_{3}$ -symmetric probability distribution on $\{(a,b,c)\in \mathbb{Z}_{{\geqslant}0}^{3}\,:\,a+b+c=n\}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,\ldots ,n\}$ with mean $n/3$ . Existence of such a distribution verifies a conjecture of Kleinberg et al. [‘The growth rate of tri-colored sum-free sets’, Discrete Anal. (2018), Paper No. 12, arXiv:1607.00047v1], which is motivated by the study of sum-free sets.

Sum-Free Sets of Integers with a Forbidden Sum

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{ 1, \ldots ,n \}$ for which the integer $2n+1$ cannot be re...

A lower bound for the k‐multicolored sum‐free problem in Zmn

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$, where $k\geq 3$ and $m\geq 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a $k$-colored sum-free set in $\mathbb{Z}_m^n$. This generalizes a result of Kleinberg-Sawin-Speyer for the case $k=3$ and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg-Sawin-Speyer. Both of these generalizations require several key new ideas.

On the structure of sum-free sets, 2

On the structure of sum-free sets, 2

On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least $2/5-c$, where $c$ is an absolute positive constant. As an application, we derive a stability version of Hu's Theorem [Proc. Amer. Math. Soc. 80 (1980), 711-712] about the maximum size of a union of two sum-free sets in $\{1,2,\ldots,n\}$. We then use this result to show that the number of subsets of $\{1,2,\ldots,n\}$ which can be partitioned into two sum-free sets is $\Theta(2^{4n/5})$, confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].