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Let q be an odd prime power and suppose that
$a,b\in \mathbb {F}_q$
are such that
$ab$
and
$(1{-}a)(1{-}b)$
are nonzero squares. Let
$Q_{a,b} = (\mathbb {F}_q,*)$
be the quasigroup in which the operation is defined by
$u*v=u+a(v{-}u)$
if
$v-u$
is a square, and
$u*v=u+b(v{-}u)$
if
$v-u$
is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies
$x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$
. Denote by
$\sigma (q)$
the number of
$(a,b)$
for which
$Q_{a,b}$
is maximally nonassociative. We show that there exist constants
$\alpha \approx 0.029\,08$
and
$\beta \approx 0.012\,59$
such that if
$q\equiv 1 \bmod 4$
, then
$\lim \sigma (q)/q^2 = \alpha $
, and if
$q \equiv 3 \bmod 4$
, then
$\lim \sigma (q)/q^2 = \beta $
.
We improve to nearly optimal the known asymptotic and explicit bounds for the number of
$\mathbb {F}_q$
-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.
We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.
Given
$E \subseteq \mathbb {F}_q^d \times \mathbb {F}_q^d$
, with the finite field
$\mathbb {F}_q$
of order q and the integer
$d\,\ge \, 2$
, we define the two-parameter distance set
$\Delta _{d, d}(E)=\{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}$
. Birklbauer and Iosevich [‘A two-parameter finite field Erdős–Falconer distance problem’, Bull. Hellenic Math. Soc.61 (2017), 21–30] proved that if
$|E| \gg q^{{(3d+1)}/{2}}$
, then
$ |\Delta _{d, d}(E)| = q^2$
. For
$d=2$
, they showed that if
$|E| \gg q^{{10}/{3}}$
, then
$ |\Delta _{2, 2}(E)| \gg q^2$
. In this paper, we give extensions and improvements of these results. Given the diagonal polynomial
$P(x)=\sum _{i=1}^da_ix_i^s\in \mathbb F_q[x_1,\ldots , x_d]$
, the distance induced by P over
$\mathbb {F}_q^d$
is
$\|x-y\|_s:=P(x-y)$
, with the corresponding distance set
$\Delta ^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}$
. We show that if
$|E| \gg q^{{(3d+1)}/{2}}$
, then
$ |\Delta _{d, d}^s(E)| \gg q^2$
. For
$d=2$
and the Euclidean distance, we improve the former result over prime fields by showing that
$ |\Delta _{2,2}(E)| \gg p^2$
for
$|E| \gg p^{{13}/{4}}$
.
A Sidon set is a subset of an Abelian group with the property that the sums of two distinct elements are distinct. We relate the Sidon sets constructed by Bose to affine subspaces of
$ \mathbb {F} _ {q ^ 2} $
of dimension one. We define Sidon arrays which are combinatorial objects giving a partition of the group
$\mathbb {Z}_{q ^ 2} $
as a union of Sidon sets. We also use linear recurring sequences to quickly obtain Bose-type Sidon sets without the need to use the discrete logarithm.
We prove that for any prime power
$q\notin \{3,4,5\}$
, the cubic extension
$\mathbb {F}_{q^{3}}$
of the finite field
$\mathbb {F}_{q}$
contains a primitive element
$\xi $
such that
$\xi +\xi ^{-1}$
is also primitive, and
$\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$
for any prescribed
$a\in \mathbb {F}_{q}$
. This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree
$n\ge 3$
.
We give a corrected version of our previous lower bound on the value set of binomials (Canad. Math. Bull., v.63, 2020, 187–196). The other results are not affected.
In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$-periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.
This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.
Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum
$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$
of the absolute value of the incomplete sums $(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $. In this very general context one of the most important results is the Pólya–Vinogradov bound
where $\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any $\varepsilon > 0$ there exists a large subset of $a \in \mathbb F_p^ \times $ such that for $${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have
Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density
T. W. Cusick conjectured that ct > 1/2. We have the elementary bound 0 < ct < 1; however, no bound of the form 0 < α ≤ ct or ct ≤ β < 1, valid for all t, is known. In this paper, we prove that ct > 1/2 – ε as soon as t contains sufficiently many blocks of 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod’ko (2017) and pursued by Emme and Hubert (2018).
Let n be a positive integer and let
$\mathbb{F} _{q^n}$
be the finite field with
$q^n$
elements, where q is a prime power. We introduce a natural action of the projective semilinear group
${\mathrm{P}\Gamma\mathrm{L}} (2, q^n)={\mathrm{PGL}} (2, q^n)\rtimes {\mathrm{Gal}} ({\mathbb F_{q^n}} /\mathbb{F} _q)$
on the set of monic irreducible polynomials over the finite field
$\mathbb{F} _{q^n}$
. Our main results provide information on the characterisation and number of fixed points.
Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In this paper, we introduce nonbinary Delsarte–Goethals codes of parameters
$(q^{m+1}\ ,\ q^{m(r+2)+2}\ ,\ {\frac{q-1}{q}(q^{m+1}-q^{\frac{m+1}{2}+r})})$
over a Galois field of order
$q=2^l$
, for all
$0\le r\le\frac{m-1}{2}$
, with m ≥ 3 odd, and show the connection of this construction to finite semifields.
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.
Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$. We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$-primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$, there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$. We prove that, for sufficiently large prime powers $q$, $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$-primitive elements. We also discuss the (weaker) translate property for extensions.
Such a sequence is eventually periodic and we denote by $P(n)$ the maximal period of such sequences for given odd $n$. We prove a lower bound for $P(n)$ by counting certain partitions. We then estimate the size of these partitions via the multiplicative order of two modulo $n$.
We obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$.