Anqi Li
We derive several new bounds for the problem of difference sets with local properties, such as establishing the super-linear threshold of the problem. For our proofs, we develop several new tools, including a variant of higher moment energies and a Ramsey-theoretic approach for the problem.
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We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}<1$. In particular, we find an upper bound of $O(2.84^n)$ on the maximum size of an affine cap in $GF(3)^n$.
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a genera…
The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd$(v,n)>1$. It is remarkable that all the known difference sets with gcd$(v,n)>1$ have the so-called character divisibility property. In 1997, Jungnickel and Schmidt posed the problem of constructing difference sets with gcd$(v,n)>1$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.
Recently, Feng and Xiang \cite{FX113} found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets.
In this paper we construct exponentionally many non-isomorphic skew Hadamard difference sets over an elementary abelian group of order $q^3$.
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.
In 1978, Robert Kibler at the National Security Agency in Fort Meade, Maryland published a description of all noncyclic difference sets with $k < 20$. Kibler's decision to stop his extensive computer search for difference sets at block size 19 was motivated partly by the difficult barrier at $k=20$, the difference sets with parameters $(96,20,4)$. In this paper, we announce the completion of the search for all $(96,20,4)$ difference sets, relying on the computer software GAP and the work of numerous authors over the last few decades. The difference sets and the symmetric designs they create ar…
In this note, we give a new construction of divisible difference sets in ${\Bbb Z}_{9}^n$ using Galois ring $GR(3^2,n)$ under the assumption of the existence of skew Hadamard difference sets in ${\Bbb F}_{3^n}$.
Forbidden multipliers in abelian difference sets
该研究发现固定阿贝尔群大子群的群自同构（除特定对合外）无法作为非平凡阿贝尔差集的乘子，结合乘子定理排除了部分差集存在性并推导了数值乘子群的界。
Cyclic relative difference sets and circulant weighing matrices
本文扩展了对循环相对差集的搜索，并将其应用于循环权矩阵的存在性研究。
In this paper, we study skew-symmetric $2$-$\{v;r,k;λ\}$ supplementary difference sets related to a certain class of complex spherical 2-codes. A classification of such supplementary difference sets is complete for $v \le 51$.
Using a class of permutation polynomials of $F_{3^{2h+1}}$ obtained from the Ree-Tits symplectic spreads in $PG(3,3^{2h+1})$, we construct a family of skew Hadamard difference sets in the additive group of $F_{3^{2h+1}}$. With the help of a computer, we show that these skew Hadamard difference sets are new when $h=2$ and $h=3$. We conjecture that they are always new when $h>3$. Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime po…
A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2-groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a link…
In this paper we prove non-existence of nontrivial partial difference sets in Abelian groups of order 8p^3, where p \geq 3 is a prime number.
Spread Construction for (36,15,6) Hadamard Difference Sets
本文提出一种直接的扩散构造，解释了九个群中32个不等价的(36,15,6)差集，并通过该构造的变体生成了第九个群中的三个差集。
A $(v, k, λ)$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. Symmetric designs fulfilling this property have the nice property of having minimal rank, which makes them interesting to study. Thus, SDP designs become useful in coding theory applications. We show in this paper that difference sets formed by direct product construction of difference sets whose developments have the SDP also have the SDP. We also establish a few results regarding isomorphisms in product construc…
We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.
A difference set is said to have classical parameters if $ (v,k, λ) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian difference sets with $d=4$. The only known examples of such difference sets correspond to the projective geometries $PG(3,q)$. We consider an arbitrary difference set with the parameters of $PG(3,q)$ in an abelian group and establish constraints on its structure. In particular, we discern embedded substructures.
In this note we prove the non-existence of two types of partial difference sets in Abelian groups of order 216. This finalizes the classification of parameters for which a partial difference set of size at most 100 exists in an Abelian group.
Deza Cayley graphs from difference sets
本文提出利用差集构造具有广义二面体子群的Deza Cayley图的方法。
We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold.
Olmez, in "Symmetric $1\frac{1}{2}$-Designs and $1\frac{1}{2}$-Difference Sets" (2014), introduced the concept of a partial geometric difference set (also referred to as a $1\frac{1}{2}$-design), and showed that partial geometric difference sets give partial geometric designs. Nowak et al., in "Partial Geometric Difference Families" (2014), introduced the concept of a partial difference family, and showed that these also give partial geometric designs. It was shown by Brouwer et al. in "Directed strongly regular graphs from $1\frac{1}{2}$-designs" (2012) that directed strongly regular graphs c…
Previous surveys by Baumert and Lopez and Sanchez have resolved the existence of cyclic (v,k,lambda) difference sets with k <= 150, except for six open cases. In this paper we show that four of those difference sets do not exist. We also look at the existence of difference sets with k <= 300 and cyclic Hadamard difference sets with v <= 10,000. Finally, we extend an earlier search of the second author to show that no cyclic projective planes exist with non-prime power orders up to two billion.
We give two new constructions of almost difference sets. The first is a generic construction of $(q^{2}(q+1),q(q^{2}-1),q(q^{2}-q-1),q^{2}-1)$ almost difference sets in certain groups of order $q^{2}(q+1)$ ($q$ is an odd prime power) having ($\mathbb{F}_{q},+)$ as a subgroup. The construction occurs in any group of order $p^{2}(p+1)$ ($p$ is an odd prime) having ($\mathbb{F}_{p^{2}},+)$ as an additive subgroup. This construction yields several infinite families of almost difference sets, many of which occur in nonabelian groups. The second construction yields $(4p,2p+1,p,p-1)$ almost differenc…
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.
A skew Hadamard difference set (SHDS) is a difference set that satisfies the skew condition. It is known that if a group G admits a skew hadamard difference set, then G is a p-group with order congruent to 3 modulo 4. We will generalize skew Hadamard difference sets to Generalized Skew Hadamard Difference Sets (GSHDS) to cover the case when p is congruent to 1 module 4, and we will extend all known results that yield necessary existence conditions of skew Hadamard difference sets to our generalization, including the known exponent bounds. We will also show a set of necessary existence conditio…
In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.
We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.
We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic classes in finite fields.
Recent progress on bounds for sets with no three terms in arithmetic progression
本文梳理无三项等差数列集合界的最新突破，介绍 Bloom–Sisask 及 Croot–Lev–Pach、Ellenberg–Gijswijt 的相关工作。
Relative Difference sets from Almost Perfect Nonlinear Functions
本文研究了几乎完美非线性函数与相对差集之间的关系，并通过Pot的结果揭示了APN函数与弯曲函数的联系。
Partial difference sets with parameters $(v,k,λ,μ)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference set in an abelian group $G$ of an order not a prime power, then $|G|=n^4$ or $9n^4$, where $n>1$ is an odd integer. In 2010, Polhill \cite{Polhill} constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using non-zero squares of a finite field, we completely answer the following…
Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater th…
The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images of difference sets in quotient groups of the original group, and searching their preimages. In this way, the search space can be dramatically decreased, and searches of groups of relatively large order (such as order 64 or order 96) can be completed.
Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that $$\lim_{n \to \infty} \frac{\log f_{s,k}(n)}{\log n} = 2,$$ which answers an old question of Erdős. In fact, we prove upper and lower bounds for $f_{s,k}(n)$ which show that its growth is closely related to the bounds in Szemerédi's theorem.
In this paper, we present constructions of abelian Paley type group schemes by using multiplicative characters of finite fields and Arasu-Dillon-Player difference sets. The constructions produce many new Paley type group schemes that were previous unknown in our classification of Paley type group schemes in finite fields of small orders.
This paper addresses the question whether triple arrays can be constructed from Youden squares developed from difference sets. We prove that if the difference set is abelian, then having $-1$ as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We also give an alternative and more direct version of the construction, leaving out the intermediate step via Youden squares. This is used when we analyse the case of non-abelian difference sets, for which we prove a sufficient condition for givi…
We generalize the construction of affine polar graphs in two different ways to obtain new partial difference sets and amorphic association schemes. The first generalization uses a combination of quadratic forms and uniform cyclotomy. In the second generalization we replace the quadratic form in the affine polar graph construction by higher degree homogeneous functions that are $p$-ary weakly regular bent. The negative Latin square type partial difference sets arising from the first generalization are new.
A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals imp…
In this paper we study a certain generalization of combinatorial designs related to almost difference sets, namely the $t$-adesign, which was coined by Cunsheng Ding in 2015. It is clear that $2$-adesigns are a kind of partially balanced incomplete block design which naturally arise in many combinatorial and statistical problems. We discuss some of their basic properties and give several constructions of $2$-adesigns (some of which correspond to new almost difference sets, and others of which correspond to new almost difference families), as well as two constructions of $3$-adesigns. We also d…
Non-disjoint strong external difference families can have any number of sets
本文通过将不相交性条件替换为非不相交性，构造出具有任意数量集合的阿贝尔强外部差族，从而绕过了经典SEDF关于多于2个集合的唯一性猜想，并揭示了其与经典SEDF的差异、联系及新的编码应用来源。
Let $G$ be an additive group of order $v$. A $k$-element subset $D$ of $G$ is called a $(v, k, λ, t)$-almost difference set if the expressions $gh^{-1}$, for $g$ and $h$ in $D$, represent $t$ of the non-identity elements in $G$ exactly $λ$ times and every other non-identity element $λ+1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. This paper reviews the recent work that has been done on almost difference sets and related topics. In this survey, we try to c…
Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.
We introduce almost supplementary difference sets (ASDS). For odd $m$, certain ASDS in ${\mathbb Z}_m$ that have amicable incidence matrices are equivalent to quaternary sequences of odd length $m$ with optimal autocorrelation. As one consequence, if $2m-1$ is a prime power, or $m \equiv 1 \mod 4$ is prime, then ASDS of this kind exist. We also explore connections to optimal binary sequences and group cohomology.
Cyclotomic skew partial difference sets and partial difference families
本文研究了循环理论在构造斜部分差集中的应用，提出了新的构造方法，并探讨了斜部分差集与部分差族之间的关系。
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.
New constructions of signed difference sets
该研究针对带符号差集的构造问题，利用部分差集、乘积方法与分圆类推导得到新的存在性结果，为其在通信与编码理论中的应用提供理论支撑。
We provide further explanation of the significance of a construction in a recent paper of Wolf [Israel J. Math. 179 (2010), 253-278] in the context of the problem of finding large subspaces in sumsets.
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang \cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
A $(v,k,λ)$ difference set in a group $G$ of order $v$ is a subset $\{d_1, d_2, \ldots,d_k\}$ of $G$ such that $D=\sum d_i$ in the group ring $\mathbb{Z}[G]$ satisfies $$D D^{-1} = n + λG,$$ where $n=k-λ$. If $D=\sum s_i d_i$, where the $s_i \in \{ \pm 1\}$, satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all $s_i=1$) and circulant weighing matrices ($G$ cyclic and $λ=0$). We will show that there are other cases of interest, and give some results on their existence.
Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the multiplicative cyclic subgroup $C$ of index twelve nor $C\cup \{0\}$ forms an almost difference set, but a union of cosets of $C$ provides us an almost difference set. We also calculate the intersection numbers and character tables of cyclotomic association schemes of class two, three and four obtained from these fields.
We confirm a conjecture of Cun Sheng Ding~\cite{Ding-Discrete} claiming that the punctured value-sets of a list of eleven trinomials over odd-degree extensions of the binary field give rise to difference sets with Singer parameters. In the course of confirming the conjecture, we show that these trinomials share the remarkable property that every element of the value-set of each trinomial has either one or four preimages. We also give a partial resolution of another conjecture of Cun Sheng Ding~\cite{Ding-Discrete} claiming that linear codes constructed from those eleven trinomials are tri-weig…
Nonexistence results and constructions for signed difference sets
该研究针对有限阿贝尔群中的符号差集，建立了新的非存在性准则与存在性构造，排除了大量开放案例并解决了部分开放问题。
We revisit the problem of constructing Menon-Hadamard difference sets. In 1997, Wilson and Xiang gave a general framework for constructing Menon-Hadamard difference sets by using a combination of a spread and four projective sets of type Q in ${\mathrm{PG}}(3,q)$. They also found examples of suitable spreads and projective sets of type Q for $q=5,13,17$. Subsequently, Chen (1997) succeeded in finding a spread and four projective sets of type Q in ${\mathrm{PG}}(3,q)$ satisfying the conditions in the Wilson-Xiang construction for all odd prime powers $q$. Thus, he showed that there exists a Men…
Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of $2$-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining $2$-groups. We…
Skew Hadamard difference sets are an interesting topic of study for over seventy years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in $\mathbb{F}_q$ where $q \equiv 3 \bmod{4}$) were the only example in abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of $\mathcal{D}_5(x^2,u)$ is a new skew Hadamard difference set in $(\mathbb{F}_{3^m},+)$ with $m$ odd, where $\mathcal{D}_n(x,u)$ denotes the first kind of Dickson polynomials of order $n$ and $u \in \mathbb{F}_q^*$.…
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first examp…
Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset $A$ of $\{1,\dots,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots,x+P_m(y)$ has size $|A|\ll N/(\log\log{N})^{c_{P_1,\dots,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.
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.
We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension.
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