A sum-product estimate in finite fields, and applications

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

A Szemerédi–Trotter type theorem, sum-product estimates in finite quasifields, and related results

Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp. The famous sum-product problem of Erdős and Szemeredi asks the following. Let $A$ be a set of $n$ real numbers. Define the _sumset_ $A+A$ of $A$ to be the set $\{x+y:x,y\in A\}$ and the _product set_ $A.A$ to be the set $\{xy:x,y\in A\}$. Must at least one of $A+A$ and $A.A$ have size $n^{2-o(1)}$? Since neither set can have size greater than $n^2$, this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was $n^{4/3}$, up to logarithmic factors [3], though the exponent has now been very slightly improved [2]. More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if $A$ is a subset of a finite field ${\mathbb F}_p$ and $p^{\delta} < |A| < p^{1-\delta}$, then one of $A+A$ and $A.A$ must have size at least $n^{1+\epsilon}$ for some $\epsilon$ that depends on $\delta$ only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that $\mathbb F_p$ does not. This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if $A$ has a large difference set $D=A-A$, it is perfectly possible for $D+D$ to be much bigger than $D$. Nevertheless, when $A$ is a set of real numbers, it still implies, as the main result of the paper shows, that the product set $D.D$ has size at least $|D|^{1+c}$, where $c>0$ is an absolute constant. The same is true of the quotient set $D/D$. The paper also proves a result of a similar kind in the finite field $\mathbb F_p$ provided that $D$ is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup $H$ of $\mathbb F_p$ that is not too large is not the difference set of any subset of $\mathbb F_p$. (The precise assumption required is that $H$ has order at most $p^{4/5-\epsilon}$.) [1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343) [2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771) [3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040) [Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)

The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide new L2→Lr extension estimates for paraboloids in certain odd dimensions with −1 non-square, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using the first association scheme graph to analyze energy sets, and as a consequence, we obtain new Lp→L4 extension theorems for spheres of primitive radii in odd dimensions, which break the Stein-Tomas result toward Lp→L4 which has stood for more than ten years. Most significantly, it follows from the results for spheres that there exists a different extension phenomenon between spheres and paraboloids in odd dimensions, namely, the Lp→L4 estimates for spheres with primitive radii are much stronger than those for paraboloids. The second purpose is to show that there is a connection between the restriction conjecture associated to paraboloids and the Erdős-Falconer distance conjecture over finite fields. The last is to prove that the Erdős-Falconer distance conjecture holds in odd dimensional spaces when we study distances between two sets: one set lies on a variety (a paraboloid or a sphere), and the other set is arbitrary in vector spaces over finite fields.

Inspired by the works of L. Carlitz [Acta Arith. 5 (1959), pp. 293–308] and Z.-W. Sun [Finite Fields Appl. 56 (2019), pp. 285–307] on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let q = p n q=p^n be an odd prime power with p p prime and n ∈ Z + n\in \mathbb {Z}^+ . Let ζ p = e 2 π i / p \zeta _p=e^{2\pi \mathbf {i}/p} and let χ \chi be a generator of the group of all multiplicative characters of the finite field F q \mathbb {F}_q . For the Gauss sum G q ( χ r ) = ∑ x ∈ F q χ r ( x ) ζ p T r F q / F p ( x ) , \begin{equation*} G_q(\chi ^{r})=\sum _{x\in \mathbb {F}_q}\chi ^{r}(x)\zeta _p^{\mathrm {Tr}_{\mathbb {F}_q/\mathbb {F}_p}(x)}, \end{equation*} we prove that det [ G q ( χ 2 i + 2 j ) ] 0 ≤ i , j ≤ ( q − 3 ) / 2 = ( − 1 ) α n ( q − 1 2 ) q − 1 2 2 p n − 1 − 1 2 , \begin{equation*} \det \left [G_q(\chi ^{2i+2j})\right ]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha _n}\left (\frac {q-1}{2}\right )^{\frac {q-1}{2}}2^{\frac {p^{n-1}-1}{2}}, \end{equation*} where α n = { 1 a m p ; if n ≡ 1 ( mod 2 ) , ( p 2 + 7 ) / 8 a m p ; if n ≡ 0 ( mod 2 ) . \begin{equation*} \alpha _n= \begin {cases} 1 &amp; \text {if } n\equiv 1\pmod 2,\\ (p^2+7)/8 &amp; \text {if } n\equiv 0\pmod 2. \end{cases} \end{equation*}

One of the most celebrated applications of Gauss’s 2 F 1 _2F_1 hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper (see M. J. Griffin, Ken Ono, N. Saikia, and W.-L. Tsai, [AGM and jellyfish swarms of elliptic curves, Amer. Math. Monthly 130 (2023), no. 4, 355-369]) in which a finite field analogue of A G M R \mathrm {AGM}_\mathbb {R} was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble jellyfish. Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such “jellyfish swarms” to prove new identities for Gauss’s class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d=22l+1 over the field F32. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

ABSTRACTHe et al. [Finite Fields and Their Applications 48 (2017), 289–305] introduced a finite field analogue for the Appell series and derived its transformations, reduction formulas as well as generating functions. In this paper, we study its other properties, which can be considered as finite field analogues for the classical one.

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

In this chapter we explore the feasibility of efficient implementation of an arithmetic processor for performing elliptic curve computations over finite fields. For a secure system, it is evident from the results of Chapter 5 that the curve and underlying field should be judiciously chosen. However we should point out that for a given underlying field there are a large number of suitable elliptic curve to choose from. If the logarithm problem in supersingular curves is indeed as hard as the logarithm problem in finite fields, and/or if the logarithm problem in non-supersingular curves is indeed intractable, then the systems discussed here are both efficient and secure and very attractive for practical usage.

The complexity of linear problems in fields

We present a deterministic $2^{O(t)}q^{\frac{t-2}{t-1}+o(1)}$ algorithm to decide whether a univariate polynomial $f$, with $t$ monomial terms and degree $<q$, has a root in the finite field $\mathbb{F}_q$. Our method is the first with complexity sublinear in $q$ when $t$ is fixed. We also prove a structural property for the nonzero roots in $\mathbb{F}_q$ of any $t$-nomial: The nonzero roots always admit a partition into no more than $2(q-1)^{\frac{t-2}{t-1}}$ cosets, each associated with one of two subgroups $S_1\subseteq S_2$ of $\mathbb{F}^*_q$. This can be thought of as a finite field analogue of Descartes' rule. A corollary of our results is the first deterministic sublinear algorithm for detecting common degree-one factors of $k$-tuples of $t$-nomials in $\mathbb{F}_q[x]$ when $k$ and $t$ are fixed. When $t$ is not fixed we show that, for $p$ prime, detecting roots in $\mathbb{F}_p$ for $f$ is $\mathbf{NP}$-hard with respect to $\mathbf{BPP}$-reductions. Finally, we prove that if the complexity of root detection is sublinear (in a refined sense), relative to the straight-line program encoding, then $\mathbf{NEXP}\nsubseteq\mathbf{P/poly}$.

Sparse univariate polynomials with many roots over finite fields

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Let $\mathbb{J}_q^k[t]$ denote the additive closure of the set of $k$th powers in the polynomial ring $\mathbb{F}_q[t]$, defined over the finite field $\mathbb{F}_q$ having $q$ elements. We show that when $s\ge k+1$ and $q \ge k^{2k+2}$, then every polynomial in $\mathbb{J}_q^k[t]$ is the sum of at most $s$ $k$th powers of polynomials from $\mathbb{F}_q[t]$. When $k$ is large and $s \ge (\frac{4}{3}+o(1)) k\log k$, the same conclusion holds without restriction on $q$. Refinements are offered that depend on the characteristic of $\mathbb{F}_q$.