Conical Kakeya and Nikodym Sets in Finite Fields

TL;DR

This paper extends Dvir's polynomial method to establish size lower bounds for Kakeya and Nikodym sets in finite fields involving conics like parabolae and hyperbolae, providing explicit bounds and constants.

Contribution

It generalizes the finite field Kakeya problem to conic sets, including explicit bounds and constants, and introduces bounds for conical Nikodym sets.

Findings

Lower bounds for conical Kakeya sets with explicit constants

Lower bounds for conical Nikodym sets including ellipses

Extension of polynomial method to non-degenerate conics in finite fields

Abstract

A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.) Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

Equations

$$Q(X,Y)=A{X}^2+BXY+C{Y}^2+DX+EY+F,A,B,C,D,E,F\in {\mathbb{F}}_q,$$

$$\left(\begin{array}[]{c}X\\ Y\end{array}\right)\mapsto\left(\begin{array}[]{c}aX+bY+e\\ cX+dY+f\end{array}\right),\quad ad\neq bc,$$

$$(x(t),y(t))=(t,m{t}^2),t\in {\mathbb{F}}_q,$$

$$(x(t),y(t))=(t,t^{-1}),t\in {\mathbb{F}}_q^{\ast },$$

$$\mathcal{H}=\{\underline{a}+t\underline{b}+t^{-1}\underline{c}:t\in {\mathbb{F}}_q^{\ast }\}.$$

$$\mathcal{P}=\{\underline{a}+t\underline{b}+t^2\underline{c}:t\in {\mathbb{F}}_q\}.$$

$$\mathcal{E}=\{\underline{a}+x(t)\underline{b}+y(t)\underline{c}:{{\color[rgb]{0,0,0}t\in\mathbb{F}_{q^{2}},t^{q+1}=1}}\}$$

$$|\mathcal{S}|\ge {\left(\frac{q-1}{2n}\right)}^n.$$

$$|\mathcal{E}|={{\color[rgb]{0,0,0}q+1}}.$$

$$v^{2}={{\color[rgb]{0,0,0}g}}u^{2}+k$$

$$s(t)$$

$$r(t)$$

$$\mathcal{E}=\{(x(t),y(t)):t\in {\mathbb{F}}_{q^2}^{\ast },t^{q+1}=1\}$$

$$x(t)$$

$$y(t)$$

$$|\mathcal{N}|\ge {\left(\frac{q-1}{2n}\right)}^n.$$

$$\mathcal{P}=\{{{\color[rgb]{0,0,0}\underline{a}}}+t\underline{b}+t^{2}\underline{c}:t\in\mathbb{F}_{q}\}.$$

$$\mathcal{H}=\{\underline{a}+t\underline{b}+t^{-1}\underline{c}:t\in {\mathbb{F}}_q^{\ast }\}.$$

$$\mathcal{E}=\{\underline{a}+\underline{b}x(t)+\underline{c}y(t):{{\color[rgb]{0,0,0}t\in\mathbb{F}_{q^{2}}^{*},t^{q+1}=1}}\}$$

$$|\mathcal{K}|\ge {\left(\frac{q-1}{2n}\right)}^n.$$

$$f=f_d+g,\mathrm{deg}(f_d)=d,\mathrm{deg}(g)<d,f_d(x_1,\dots ,x_n)=\sum _{i_1+\dots +i_n=d}{e}_{i_1,\dots ,i_n}{x}_1^{i_1}{x}_2^{i_2}\cdots x_n^{i_n}.$$

$$\mathcal{P}=\{\underline{a}+t\underline{b}+t^2\underline{c}:t\in {\mathbb{F}}_q\}.$$

$$f_d(\underline{a}+\underline{b}t+\underline{c}{t}^2)$$

$$=\sum _{i_1+\dots +i_n=d}{e}_{i_1,\dots ,i_n}(a_1+b_{1}t+c_1{t}^2{)}^{i_1}(a_2+b_{2}t+c_2{t}^2{)}^{i_2}\cdots (a_n+b_{n}t+c_n{t}^2{)}^{i_n}.$$

$$\sum _{i_1+\dots +i_n=d}{e}_{i_1,\dots ,i_n}{c}_1^{i_1}{c}_2^{i_2}\cdots c_n^{i_n}{t}^{2d}$$

$$=t^{2d}{f}_d(\underline{c})$$

$$\mathcal{H}=\{\underline{a}+t\underline{b}+t^{-1}\underline{c}:t\in {\mathbb{F}}_q^{\ast }\}.$$

$$f_d(\underline{a}+\underline{b}t+\underline{c}{t}^{-1})$$

$$\sum _{i_1+\dots +i_n=d}{e}_{i_1,\dots ,i_n}{c}_1^{i_1}\cdots c_n^{i_n}=f_d(\underline{c}).$$

$$(f_d{)}^{\underline{i}}(\underline{x})=(f^{\underline{i}}{)}_{d^{\prime }}(\underline{x})$$

$$\text{Mult}(f^{\underline{i}},\underline{a})\ge \text{Mult}(f,\underline{a})-wt(\underline{i}).$$

$$\text{Mult}(f(g(\underline{x})),\underline{a})\ge \text{Mult}(f(\underline{x}),g(\underline{a})).$$

Full text

Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria, E-mail: {audie.warren,arne.winterhof}@oeaw.ac.at

Keywords. Kakeya set, Nikodym set, polynomial method, method of multiplicities, conic

1 Introduction

A subset $\mathcal{K}\subseteq\mathbb{F}_{q}^{n}$ of $n$-dimensional vectors over the finite field $\mathbb{F}_{q}$ of $q$ elements is called a Kakeya set in $\mathbb{F}_{q}^{n}$ if it contains a line in each direction. Using the polynomial method Dvir [3, Theorem 1.5] showed that any Kakeya set in $\mathbb{F}_{q}^{n}$ contains at least $c_{n}q^{n}$ elements with a constant $c_{n}$ depending only on $n$, see also [7, Theorem 2.11].

A set $\mathcal{N}\subseteq\mathbb{F}_{q}^{n}$ is called a Nikodym set in $\mathbb{F}_{q}^{n}$ if for each point $x\in\mathbb{F}_{q}^{n}$ there is a line $L$ containing $x$ such that $L\setminus\{x\}\subseteq\mathcal{N}$. A small variation of Dvir’s proof also provides that any Nikodym set in $\mathbb{F}_{q}^{n}$ contains at least $c_{n}q^{n}$ elements with a constant $c_{n}$ depending only on $n$, see [7, Theorem 2.9].

Now let $q$ be the power of an odd prime. The set of zeros $(x,y)\in\mathbb{F}_{q}^{2}$ of a polynomial

[TABLE]

of degree $2$, that is $A$, $B$ and $C$ are not all zero, is called a conic in $\mathbb{F}_{q}^{2}$. In the degenerate case, that is $Q(X,Y)$ is reducible over the algebraic closure of $\mathbb{F}_{q}$, we get a pair of (intersecting, parallel or identical) lines, a point or the empty set. We restrict ourselves to the non-degenerate case, that is, $Q(X,Y)$ is absolutely irreducible over $\mathbb{F}_{q}$, since the degenerate case is either trivial or can be reduced to the previously studied case of a single line. We may assume $C=1$, $C=0$ and $A=1$, or $A=C=0$ and $B=1$. After regular affine substitutions

[TABLE]

we are left with the following cases where $g$ is any fixed non-square in $\mathbb{F}_{q}^{*}$:

• •

$A=C=0$, $B=1$: hyperbola $\{(x,y)\in\mathbb{F}_{q}^{2}:x\neq 0\mbox{ and }y=x^{-1}\}=\{(t,t^{-1})\in\mathbb{F}_{q}^{2}:t\in\mathbb{F}_{q}^{*}\}$.

• •

$(A,C)\in\{(1,0),(0,1)\}$: parabola $\{(x,y)\in\mathbb{F}_{q}^{2}:y=mx^{2}\}=\{(t,mt^{2}):t\in\mathbb{F}_{q}\}$, where $m\in\{1,g\}$.

• •

$C=1$, $A\neq 0$: ellipse \{(x,y)\in\mathbb{F}_{q}^{2}:y^{2}={{\color[rgb]{0,0,0}g}}x^{2}+k\}, $k\in\mathbb{F}_{q}^{*}$.

(Note that conics defined by $Y^{2}=X^{2}+k$ can be transformed into the form $XY=1$ and are hyperbolae.)

For parabolae and hyperbolae the parametrisations

[TABLE]

and

[TABLE]

respectively, are obvious. However, we can also derive parametrisations of ellipses $(x(t),y(t))$ where $t\in\mathbb{F}_{q^{2}}$ with $t^{q+1}=1$, see Section 2 below.

To extend the definition of a conic to a general dimension $n\geq 2$, we embed any conic in $\mathbb{F}_{q}^{2}$ into a plane in $\mathbb{F}_{q}^{n}$. That is for some vectors $\underline{a},\underline{b},\underline{c}\in\mathbb{F}_{q}^{n}$ where $\underline{b}$ and $\underline{c}$ are linearly independent:

• •

A(n embedding of a) hyperbola in $\mathbb{F}_{q}^{n}$ is a set

[TABLE]

• •

A(n embedding of a) parabola in $\mathbb{F}_{q}^{n}$ is a set

[TABLE]

• •

An (embedding of an) ellipse in $\mathbb{F}_{q}^{n}$ is a set

[TABLE]

where (x(t),y(t)){{\color[rgb]{0,0,0}\in\mathbb{F}_{q}^{2}}} is given in Section 2.

(Without the linear independence of $\underline{b}$ and $\underline{c}$ the embedding can have fewer points than the embedded conic. Hence, a hyperbola has $q-1$ points, a parabola $q$ points and an ellipse $q+1$ points.)

We give adaptations of Dvir’s proof to give bounds on conical Kakeya and Nikodym sets defined as follows.

A subset $\mathcal{N}\subseteq\mathbb{F}_{q}^{n}$ is called a conical Nikodym set if for all $\underline{x}\in\mathbb{F}_{q}^{n}$ there is a non-degenerate conic $\mathcal{C}$ of the form $(\ref{H})$, $(\ref{P})$ or $(\ref{E})$ with $\underline{x}\in\mathcal{C}$ and $\mathcal{C}\setminus\{\underline{x}\}\subseteq\mathcal{N}$.

In order to define conical Kakeya sets, we must decide on how to define the ’direction’ of a conic which can be identified with the ’point(s) at infinity’ of the conic, that is, a hyperbola has two directions $\underline{b}$ and $\underline{c}$, a parabola has one direction $\underline{c}$, and an ellipse has no direction.

A subset $\mathcal{K}\subseteq\mathbb{F}_{q}^{n}$ is called a conical Kakeya set if for all $\underline{d}\in\mathbb{F}_{q}^{n}\setminus\{\underline{0}\}$

there exist \underline{a},\underline{b},{{\color[rgb]{0,0,0}\underline{c}}}\in\mathbb{F}_{q}^{n} such that $\underline{b}$ and $\underline{c}$ are linearly independent and there is a conic contained in $\mathcal{K}$ either of the form $(\ref{H})$ with $\underline{d}\in\{\underline{b},\underline{c}\}$ or of the form $(\ref{P})$ with $\underline{d}=\underline{c}$.

We prove the following Theorem.

Theorem 1**.**

Let $\mathcal{S}\subseteq\mathbb{F}_{q}^{n}$ with $n\geq 2$ be a conical Kakeya or Nikodym set, where $q$ is a power of an odd prime. Then

[TABLE]

For conical Kakeya sets the lower bound $c_{n}q^{n}$ with a constant depending on $n$ follows from [5, Corollary 1.10]. However, in contrast to [5] our constant is explicit and in Section 5 we use the method of multiplicities of [4] to determine a constant of the form $c_{n}=c^{n}$ where $c$ does not depend on $n$.

Moreover, at the end of the paper we give an example of a subset of $\mathbb{F}_{q}^{2}$ of size $q+1$ which contains for each $\underline{c}$, resp., $\underline{b}$ an ellipse of form $(\ref{E})$. Hence, it is necessary to exclude ellipses in the definition of conical Kakeya sets.

In Section 2 we derive a parametrisation for ellipses needed in the proof of Theorem 1. In Section 3 we prove Theorem 1 for conical Nikodym sets and in Section 4 for conical Kakeya sets. In Section 5 we improve the constant $c_{n}$ using the method of multiplicities. In Section 6 we conclude with some final remarks.

For readers not familiar with the polynomial method we refer to the book of Guth [7] and the survey article of Tao [13] as excellent starting points.

2 Parametrisation of ellipses

In this section we derive a parametrisation for ellipses, which is vital in our proof of Theorem 1 for elliptic Nikodym sets.

Consider the ellipse \mathcal{E}=\{(x,y)\in\mathbb{F}_{q}^{2}:y^{2}={{\color[rgb]{0,0,0}g}}x^{2}+k\}, where $g$ is a non-square in $\mathbb{F}_{q}^{*}$ and $k\in\mathbb{F}_{q}^{*}$. By [8, Lemma 6.24] we have

[TABLE]

Using analogues $s(t)$ and $r(t)$ of sine and cosine for finite fields defined below, see for example [9, Definition 15.5], we are able to find parametrisations of ellipses.

Note that a solution $z$ of $z^{2}=g$ is not an element of $\mathbb{F}_{q}$: $z\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$. Let $(u,v)\in\mathbb{F}_{q}^{2}$ be any fixed solution of

[TABLE]

which exists by $(\ref{nop})$. Then verify that

[TABLE]

is a solution of $s(t)^{2}=g(r(t)^{2}-1)$. It can be easily checked that

[TABLE]

with

[TABLE]

Since $r(t)^{q}=r(t)$ and $s(t)^{q}=s(t)$ (using z^{q}=z{{\color[rgb]{0,0,0}g}}^{(q-1)/2}=-z because {{\color[rgb]{0,0,0}g}} is a non-square in $\mathbb{F}_{q}^{*}$) we have $(r(t),s(t))\in\mathbb{F}_{q}^{2}$, so that $(x(t),y(t))\in\mathbb{F}_{q}^{2}$.

3 Conical Nikodym sets

In this section we prove Theorem 1 for conical Nikodym sets.

Proposition 1**.**

Let $\mathcal{N}\subseteq\mathbb{F}_{q}^{n}$ with $n\geq 2$ and $q$ the power of an odd prime be a conical Nikodym set. Then we have

[TABLE]

Proof.

Suppose $|\mathcal{N}|<\left(\frac{q-1}{2n}\right)^{n}$. By [7, Lemma 2.4], there is a non-zero polynomial $f$ with f({{\color[rgb]{0,0,0}\underline{s}}})=0 for all {{\color[rgb]{0,0,0}\underline{s}}}\in\mathcal{N}, and $\deg(f)\leq n|\mathcal{N}|^{1/n}\leq\frac{q-3}{2}$. Take any $\underline{x}\in\mathbb{F}_{q}^{n}$. As $\mathcal{N}$ is conical Nikodym, there exists a conic $\mathcal{C}$ of the form $(\ref{H})$, $(\ref{P})$ or $(\ref{E})$ with $\underline{x}\in\mathcal{C}$ and $\mathcal{C}\setminus\{\underline{x}\}\subseteq\mathcal{N}$. We split into cases depending on the form of the conic $\mathcal{C}$.

Firstly assume the conic $\mathcal{C}$ is a parabola $\mathcal{P}$. Parametrise this parabola as

[TABLE]

Applying these points to the polynomial $f$, we define $F(t)=f(\underline{a}+t\underline{b}+t^{2}\underline{c})$ a univariate polynomial in $t$ of degree $\deg(F)\leq q-3$. We also know that it has $q-1$ zeros corresponding to the points of the parabola lying in $\mathcal{N}$, and thus must be zero on the whole parabola, in particular $f(\underline{x})=0$.

Secondly assume $\mathcal{C}$ is a hyperbola $\mathcal{H}$. Parametrise this hyperbola as

[TABLE]

Applying these points to the polynomial $f$, we define $F(t)=t^{\deg(f)}f(\underline{a}+t\underline{b}+t^{-1}\underline{c})$ a univariate polynomial in $t$ of degree $\deg(F)\leq q-3$. We also know that it has $q-2$ zeros corresponding to the points of the hyperbola $\mathcal{H}$ lying in $\mathcal{N}$, and thus must be zero on the whole hyperbola. Again we find $f(\underline{x})=0$.

Thirdly we assume that $\mathcal{C}$ is an ellipse $\mathcal{E}$. The number of points on this ellipse is $q+1$. We use our parametrisation of an ellipse; it has form

[TABLE]

for some appropriate choice of $\underline{a}$, $\underline{b}$ and $\underline{c}$, and $(x(t),y(t))$ are given in Section 2. We consider the polynomial $F(t)=t^{\deg(f)}f(\underline{a}+\underline{b}x(t)+\underline{c}y(t))$. This polynomial is univariate in $t$ of degree $\deg(F)\leq q-3$. We know that it has $q$ zeros (in $\mathbb{F}_{q^{2}}$) corresponding to the points of the ellipse $\mathcal{E}$ lying in $\mathcal{N}$, and thus must be zero on the whole ellipse. We again find that $f(\underline{x})=0$.

In all three cases we found that $f(\underline{x})=0$. As $\underline{x}$ was chosen arbitrarily we conclude that $f(\underline{x})=0$ for all $\underline{x}\in\mathbb{F}_{q}^{n}$. As $\deg(f)\leq\frac{q-3}{2}$ the polynomial $f$ must be the zero polynomial, a contradiction. ∎

4 Conical Kakeya sets

In this section we prove Theorem 1 for conical Kakeya sets.

Proposition 2**.**

Let $\mathcal{K}\subseteq\mathbb{F}_{q}^{n}$ with $n\geq 2$ and $q$ the power of an odd prime be a conical Kakeya set. Then

[TABLE]

Proof.

Suppose that $|\mathcal{K}|<\left(\frac{q-1}{2n}\right)^{n}$. By [7, Lemma 2.4] there exists $f$ a non-zero polynomial with f({{\color[rgb]{0,0,0}\underline{s}}})=0\ for all {{\color[rgb]{0,0,0}\underline{s}}}\in\mathcal{K}, with degree $d\leq\frac{q-3}{2}$. We split this polynomial into a sum of its greatest degree part and the lower degree terms as

[TABLE]

Note that as $f_{d}$ is homogeneous, $f_{d}(\underline{0})=0$. Take any $\underline{x}\in\mathbb{F}_{q}^{n}\setminus\{\underline{0}\}$. As $\mathcal{K}$ is conical Kakeya, there exists some conic $\mathcal{C}$ of the form $(\ref{H})$ or $(\ref{P})$ with $\underline{x}$ appearing as $\underline{c}$ for parabolae and $\underline{b}$ or $\underline{c}$ for hyperbolae from Section 1. We split into two cases depending on which type of conic $\mathcal{C}$ defines.

First assume $\mathcal{C}$ is a parabola $\mathcal{P}$. It has parametrisation

[TABLE]

We consider the polynomial F(t){{\color[rgb]{0,0,0}=}}f(\underline{a}+t\underline{b}+t^{2}\underline{c}), which is univariate in $t$ of degree $2d$. Since $f$ is zero on $\mathcal{K}$, $F(t)=0$ for all $t\in\mathbb{F}_{q}$. Then as $\deg(F)=2d<q$, $F$ is identically zero. We note that the coefficient of $t^{2d}$ in $F(t)$ is the coefficient of $t^{2d}$ in $f_{d}(\underline{a}+\underline{b}t+\underline{c}t^{2})$:

[TABLE]

Upon multiplying out to find the coefficient of $t^{2d}$ we have

[TABLE]

and thus as $F$ is identically zero, $f_{d}(\underline{c})=f_{d}(\underline{x})=0$.

Secondly we assume $\mathcal{C}$ is a hyperbola $\mathcal{H}$. Up to the relabelling of $t\rightarrow t^{-1}$, we may assume it has parametrisation

[TABLE]

Consider the univariate polynomial $F(t)=t^{d}f(\underline{a}+t\underline{b}+t^{-1}\underline{c})$, which is of degree $\deg(F)=2d<q-1$. The polynomial $f$ vanishes on $\mathcal{K}$, and so $F(t)=0$ for all $t\in\mathbb{F}_{q}^{*}$. As $\deg(F)<q-1$ with $F$ having at least $q-1$ zeros, we have that $F(t)$ is identically zero, in particular its constant term is zero. We calculate the constant term of $F(t)$; it is precisely the coefficient of $t^{-d}$ in $f_{d}(\underline{a}+\underline{b}t+\underline{c}t^{-1})$,

[TABLE]

so the coefficient of $t^{-d}$ is

[TABLE]

Thus $f_{d}(\underline{c})=f_{d}(\underline{x})=0$.

In both cases we have $f_{d}(\underline{x})=0$. Since we already knew that $f_{d}(\underline{0})=0$, we have $f_{d}(\underline{x})=0$ for all $\underline{x}\in\mathbb{F}_{q}^{n}$. As $d<q$, $f_{d}$ is identically zero, which is a contradiction. ∎

5 Improvements via the method of multiplicities

The ’method of multiplicities’ was used in [4], see also [13], to prove a constant of $2^{-n}$ for line Kakeya sets. This involves Hasse derivatives and exploiting polynomials which vanish to a high multiplicity on a particular set.

Let $\underline{x}=(x_{1},...,x_{n})$ and $f\in\mathbb{F}_{q}[\underline{x}]$. For a vector $\underline{i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n}$, the $\underline{i}$’th Hasse derivative of $f$, which we denote $f^{\underline{i}}(\underline{x})$, is the coefficient of $\underline{y}^{\underline{i}}$ in the polynomial $f(\underline{x}+\underline{y})$, where $\underline{y}^{\underline{i}}$ is the monomial $y_{1}^{i_{1}}y_{2}^{i_{2}}...y_{n}^{i_{n}}$.

For $f\in\mathbb{F}_{q}[\underline{x}]$ and $\underline{a}\in\mathbb{F}_{q}^{n}$, the multiplicity of $f$ at $\underline{a}$, denoted $\text{Mult}(f,\underline{a})$, is the largest integer $M$ such that for all vectors $\underline{i}\in\mathbb{N}^{n}$ of weight $wt(\underline{i})<M$, the $\underline{i}$’th Hasse derivative of $f$ is zero at $\underline{a}$, that is, $f^{\underline{i}}(\underline{a})=0$, where $wt(\underline{i})=i_{1}+\ldots+i_{n}$.

We make use of five results relating to multiplicities and Hasse derivatives. These results, with proofs, can be found in [4], see also [13].

Lemma 1**.**

Hasse derivatives ’commute’ with taking homogeneous parts of highest degree. That is, for $f\in\mathbb{F}_{q}[\underline{x}]$ of total degree $d$, letting $f_{d}$ denote the homogeneous part of $f$ of degree $d$, we have

[TABLE]

where $d^{\prime}\leq d-wt(\underline{i})$ is the degree of $f^{\underline{i}}$.

Lemma 2**.**

Taking $\underline{i}$’th Hasse derivatives reduces multiplicity by at most the weight of $\underline{i}$. That is,

[TABLE]

Lemma 3**.**

Multiplicities of compositions of polynomials $f(g(\underline{x}))$ at $\underline{a}$ is at least the multiplicity of $f$ at $g(\underline{a})$. That is,

[TABLE]

Lemma 4** (Vanishing lemma for multiplicities).**

Let $f\in\mathbb{F}_{q}[\underline{x}]$ be of degree $d$. Then

[TABLE]

Lemma 5**.**

Suppose $\mathcal{S}\subseteq\mathbb{F}_{q}^{n}$ such that for some natural numbers $m,d$ we have

[TABLE]

Then there is a non-zero polynomial $f\in\mathbb{F}_{q}[\underline{x}]$ of degree at most $d$, such that $\text{Mult}(f,\underline{s})\geq m$ for all $\underline{s}\in\mathcal{S}$.

Note that Lemma 5 is satisfied if $|\mathcal{S}|\leq\left(\frac{d}{m+n}\right)^{n}$.

5.1 Conical Nikodym sets

In this section we use the method of multiplicities to prove the following theorem.

Theorem 2**.**

Let $\mathcal{N}\subset\mathbb{F}_{q}^{n}$ be a conical Nikodym set, with $q$ a power of an odd prime. Then we have

[TABLE]

Proof.

We begin by taking a large multiple of $q$, call it $lq$ for some positive integer $l$, and define

[TABLE]

Assume that $|\mathcal{N}|\leq\left(\frac{lq-1}{m+n}\right)^{n}$. By Lemma 5, there is a non-zero polynomial $f\in\mathbb{F}_{q}[\underline{x}]$ of degree $d<lq$, such that $\text{Mult}(f,\underline{x})\geq m$ for all $\underline{x}\in\mathcal{N}$. Let $f_{d}(\underline{x})$ be the homogeneous part of $f$ of degree $d$, which we know is not the zero polynomial. We aim to show that $f_{d}$ has high multiplicity everywhere in $\mathbb{F}_{q}^{n}$, and thus must be the zero polynomial. Indeed, we will show it has multiplicity $l$ everywhere.

Choose $\underline{i}\in\mathbb{N}^{n}$ with $wt(\underline{i})<l$, and $\underline{z}\in\mathbb{F}_{q}^{n}$. We aim to show that $(f_{d})^{\underline{i}}(\underline{z})=0$. The case $\underline{z}=\underline{0}$ is trivial, so we assume $\underline{z}\not=\underline{0}$. As $\mathcal{N}$ is conical Nikodym, there is a conic $\mathcal{C}$ such that $\underline{z}\in\mathcal{C}$ and $\mathcal{P}\setminus\{\underline{z}\}\subset\mathcal{N}$. We split into cases depending on the conic $\mathcal{C}$, aiming to show that $f^{\underline{i}}(\underline{z})=0$.

Case 1 - Parabola

Assume $\mathcal{C}$ is a parabola, which we can parametrise as $\underline{c}t^{2}+\underline{b}t+\underline{a}$. We know by the properties of $f$ that $\text{Mult}(f,\underline{c}t^{2}+\underline{b}t+\underline{a})\geq m$ for $q-1$ values of $t$. By Lemma 2, we have $\text{Mult}(f^{\underline{i}},\underline{c}t^{2}+\underline{b}t+\underline{a})\geq m-wt(\underline{i})$. We can now use Lemma 3 to get

[TABLE]

for $q-1$ values of $t$. Note that the polynomial $f^{\underline{i}}(\underline{c}x^{2}+\underline{b}x+\underline{a})$ has degree $d^{\prime}\leq 2\deg(f^{\underline{i}})\leq 2(d-wt(\underline{i}))$. However, $f^{\underline{i}}(\underline{c}x^{2}+\underline{b}x+\underline{a})$ has multiplicity at least $m-wt(\underline{i})$ for $q-1$ values of $t$, so that by Lemma 4, as $d<lq$, $wt(\underline{i})<l$, we have

[TABLE]

so that $f^{\underline{i}}(\underline{c}x^{2}+\underline{b}x+\underline{a})$ is in fact the zero polynomial. But then $f^{\underline{i}}(\underline{z})=0$, as needed.

Case 2 - Hyperbola

Assume $\mathcal{C}$ is a hyperbola, which we can parametrise as $\underline{b}t+\underline{c}t^{-1}+\underline{a}$. We know by the properties of $f$ that $\text{Mult}(f,\underline{b}t+\underline{c}t^{-1}+\underline{a})\geq m$ for $q-2$ values of $t$. By Lemma 2, we have $\text{Mult}(f^{\underline{i}},\underline{b}t+\underline{c}t^{-1}+\underline{a})\geq m-wt(\underline{i})$. We have $d^{\prime}=\deg(f^{\underline{i}})\leq d-wt(\underline{i})$, and we define the polynomial $F(t)=t^{d^{\prime}}f^{\underline{i}}(\underline{b}t+\underline{c}t^{-1}+\underline{a})$ which has degree $2d^{\prime}$, and also has multiplicity at least $m-wt(\underline{i})$ for $q-2$ values of $t$. By the vanishing lemma, we have

[TABLE]

so that $F(t)$ is the zero polynomial. In particular, when we input the value $t_{0}\neq 0$ corresponding to $\underline{z}$ on the hyperbola, we get zero. Then $F(t_{0})=t_{0}^{d^{\prime}}f^{\underline{i}}(\underline{b}t_{0}+\underline{c}t_{0}^{-1}+\underline{a})=t_{0}^{d^{\prime}}f^{\underline{i}}(\underline{z})=0\implies f^{\underline{i}}(\underline{z})=0$ as needed.

Case 3 - Ellipse

Assume $\mathcal{C}$ is an ellipse, which we can parametrise as $\underline{b}x(t)+\underline{c}y(t)+\underline{a}$ with $\underline{b}$ and $\underline{c}$ linearly independent. We know by the properties of $f$ that $\text{Mult}(f,\underline{b}x(t)+\underline{c}y(t)+\underline{a})\geq m$ for $q$ values of $t$. By Lemma 2, we have $\text{Mult}(f^{\underline{i}},\underline{b}x(t)+\underline{c}y(t)+\underline{a})\geq m-wt(\underline{i})$. We have $d^{\prime}=\deg(f^{\underline{i}})\leq d-wt(\underline{i})$, and we define the polynomial $F(t)=t^{d^{\prime}}f^{\underline{i}}(\underline{b}x(t)+\underline{c}y(t)+\underline{a})$ which has degree $2d^{\prime}$, and also has multiplicity at least $m-wt(\underline{i})$ for $q$ values of $t\in\mathbb{F}_{q^{2}}$. By the vanishing lemma, we have

[TABLE]

so that $F(t)$ is the zero polynomial. In particular, when we input the value $t_{0}\neq 0$ corresponding to $\underline{z}$ on the ellipse, we get zero. Then $F(t_{0})=t_{0}^{d^{\prime}}f^{\underline{i}}(\underline{b}x(t)+\underline{c}y(t)+\underline{a})=t_{0}^{d^{\prime}}f^{\underline{i}}(\underline{z})=0\implies f^{\underline{i}}(\underline{z})=0$ as needed.

This was for arbitrary $\underline{z}$, so we have $\text{Mult}(f,\underline{z})\geq l$ for all $\underline{z}\in\mathbb{F}_{q}^{n}$, and we may use the vanishing lemma a final time to show

[TABLE]

so $f$ is in fact the zero polynomial, a contradiction. We may allow $l$ to go to infinity, so that

[TABLE]

as needed. ∎

5.2 Conical Kakeya sets

In this section we adapt the proof of [4] for line Kakeya sets to conical Kakeya sets.

Theorem 3**.**

Let $\mathcal{K}\subset\mathbb{F}_{q}^{n}$ be a conical Kakeya set, with odd q. Then we have

[TABLE]

Proof.

We begin by taking a large multiple of $q$, call it $lq$, and define $m=3l$.

Assume that $|\mathcal{K}|\leq\left(\frac{lq-1}{m+n}\right)^{n}$. By Lemma 5, there is a non-zero polynomial $f\in\mathbb{F}_{q}[\underline{x}]$ of degree $d<lq$, such that $\text{Mult}(f,\underline{k})\geq m$ for all $\underline{k}\in\mathcal{K}$. Let $f_{d}$ denote the homogeneous part of $f$ with highest degree $d$. We will show that this polynomial has multiplicity $l$ everywhere, so that $f$ must be the zero polynomial.

Let $\underline{c}\in\mathbb{F}_{q}^{n}$ be arbitrary and non-zero (the zero case is trivial), and take $\underline{i}\in\mathbb{N}^{n}$ with $wt(\underline{i})<l$. As $\mathcal{K}$ is conical Kakeya, there is either a parabola, hyperbola or an ellipse with direction $\underline{c}$ contained in $\mathcal{K}$. We split into cases, with the aim to show $(f^{\underline{i}})_{d}(\underline{c})=0$.

Case 1 - Parabola Assume there is a parabola of the form $\underline{c}t^{2}+\underline{b}t+\underline{a}$ contained in $K$. We know by the properties of $f$ that $\text{Mult}(f,\underline{c}t^{2}+\underline{b}t+\underline{a})\geq m$ for $t\in\mathbb{F}_{q}$. By Lemma 2, we have $\text{Mult}(f^{\underline{i}},\underline{c}t^{2}+\underline{b}t+\underline{a})\geq m-wt(\underline{i})$. We can now use Lemma 3 to get

[TABLE]

Note that the polynomial $f^{\underline{i}}(\underline{c}t^{2}+\underline{b}t+\underline{a})$ has degree $d^{\prime}\leq 2\deg(f^{\underline{i}})\leq 2(\deg(f)-wt(\underline{i}))=2(d-wt(\underline{i}))$. However, $f^{\underline{i}}(\underline{c}t^{2}+\underline{b}t+\underline{a})$ has multiplicity at least $m-wt(\underline{i})$ everywhere in $\mathbb{F}_{q}$, so that by Lemma 4, as $d<lq$, $wt(\underline{i})<l$, we have

[TABLE]

so that $f^{\underline{i}}(\underline{c}t^{2}+\underline{b}t+\underline{a})$ is in fact the zero polynomial.

The next observation is crucial; the coefficient of $x^{2\deg(f^{\underline{i}})}$ in $f^{\underline{i}}(\underline{c}t^{2}+\underline{b}t+\underline{a})$ is precisely $(f^{\underline{i}})_{d^{\prime}}(\underline{c})$, as only this highest degree homogeneous part could reach the highest power of $x$. But then by Lemma 1, we have

[TABLE]

as needed.

Case 2 - Hyperbola Up to a relabelling of $t\rightarrow t^{-1}$, we may parametrise the hyperbola as $\underline{c}t+\underline{b}t^{-1}+\underline{a}$. As the polynomial $f(\underline{x})$ has multiplicity $m$ everywhere in $\mathcal{K}$, $\text{Mult}(f,\underline{c}t+\underline{b}t^{-1}+\underline{a})\geq m$ for $t\in\mathbb{F}_{q}^{*}$. We then have that $\text{Mult}(f^{\underline{i}},\underline{c}t+\underline{b}t^{-1}+\underline{a})\geq m-wt(\underline{i})$ for $t\in\mathbb{F}_{q}^{*}$. Let $d^{\prime}$ denote the degree of $f^{\underline{i}}(\underline{x})$. We have $d^{\prime}\leq d-wt(\underline{i})$, and we define the polynomial $F(t)=t^{d^{\prime}}f^{\underline{i}}(\underline{c}t+\underline{b}t^{-1}+\underline{a})$. Note that $F(t)$ has multiplicity at least $m-wt(\underline{i})$ for all $t\in\mathbb{F}_{q}^{*}$, so that by the vanishing lemma,

[TABLE]

Therefore $F(t)$ is the zero polynomial. In particular, its highest degree term is zero. The coefficient of $t^{2d^{\prime}}$ in $F(t)$ is precisely $(f^{\underline{i}})_{d^{\prime}}(\underline{c})$. By Lemma 1, we have $(f^{\underline{i}})_{d^{\prime}}(\underline{c})=(f_{d})^{\underline{i}}(\underline{c})=0$, as needed.

We now have that $\text{Mult}(f_{d},\underline{a})\geq l$ for all $\underline{a}\in\mathbb{F}_{q}^{n}$. We may now use Lemma 4 to show

[TABLE]

so that $f_{d}$ is the zero polynomial, a contradiction. We therefore must have $|\mathcal{K}|\geq\left(\frac{lq-1}{m+n}\right)^{n}$. As $l$ was an arbitrary large integer, we may allow $l\rightarrow\infty$, so we have

[TABLE]

as needed. ∎

6 Final remarks

• •

For line Kakeya sets Dvir gave a construction of size at most

[TABLE]

see [11, Theorem 7]. This construction can easily be adjusted to conical Kakeya sets. However, we lose a factor $2$. We explain this for parabolae. For hyperbolae and ellipses one can deal analogously. Since otherwise our result is trivial we assume $n\geq 3$. For any direction $\underline{c}=(c_{1},\ldots,c_{n})\neq\underline{0}$ we take $\underline{b}=(b_{1},\ldots,b_{n})$ with $b_{1}=1$ and $b_{i}=0$ for $i=2,\ldots,n$ if $c_{n}\neq 0$ and $(b_{1},\ldots,b_{n-1},0)$ any vector which is linearly independent to $\underline{c}$ if $c_{n}=0$. We also take $\underline{a}=(a_{1},\ldots,a_{n})$ with $a_{n}=0$. Then for $c_{n}=0$ the parabola $\underline{a}+t\underline{b}+t^{2}\underline{c}$ lies in $\mathbb{F}_{q}^{n-1}\times\{0\}$ which contains $q^{n-1}$ points. For $c_{n}\neq 0$ choose $b_{1}=1$ and $b_{i}=0$ for $i=2,\ldots,n$ and note that $\underline{b}$ and $\underline{c}$ are linearly independent. Choosing $a_{i}=c_{i}^{2}(2c_{n})^{-2}$ for $i=2,\ldots,n-1$ we see that $\underline{a}+t\underline{b}+t^{2}\underline{c}$ is of the form $(\alpha_{1},\ldots,\alpha_{n})$ with $\alpha_{i}+\alpha_{n}^{2}=a_{i}+t^{2}c_{i}+t^{4}c_{n}^{2}=(t^{2}c_{n}+c_{i}(2c_{n})^{-1})^{2}$ for $i=2,\ldots,n-1$ by the choice of $a_{i}$. Hence, the parabola lies in the set $\{(\alpha_{1},\ldots,\alpha_{n}):\alpha_{i}+\alpha_{n}^{2}\mbox{ is a square for }i=2,\ldots,n-1\mbox{ and }\alpha_{n}\neq 0\}$. We have $q-1$ choices for $\alpha_{n}$, $q$ for $\alpha_{1}$ and $(q+1)/2$ for each $\alpha_{i}$ with $i=2,\ldots,n-1$. Hence, the size of our conical Kakeya set is at most

[TABLE]

• •

In [2, Definition 6], the authors introduced Kakeya sets of degree $r$ which coincide with line Kakeya sets if $r=1$. For $r=2$ this definition differs from our definition of parabolic Kakeya sets by the condition that $\underline{b}$ and $\underline{c}$ are allowed to be linearly dependent. If $q\equiv 1\bmod(r+1)$, in Lemma 7 they also give constructions of size at most $\left(\frac{q-1}{r+1}+1\right)^{n+1}$. For $r=2$ the construction is

[TABLE]

However, to satisfy the linear independence condition we have to add lines for the directions for which $(c_{1},\ldots,c_{n})$ and $(c_{1}^{2},\ldots,c_{n}^{2})$ are linearly dependent, that is, $(c_{1},\ldots,c_{n})\in\{0,c\}^{n}$ for some $c\in\mathbb{F}_{q}^{*}$ and we have to add $O(q^{2})$ further vectors, that is, we have the upper bound

[TABLE]

It is not difficult to extend Theorem 3 to such Kakeya sets of degree $r$ (with a linear independence condition) giving the lower bound $\left(\frac{q}{r+1}\right)^{n}$. (Without the linear independence condition we can get only a weaker lower bound since the polynomial curves may contain only $\lceil q/r\rceil$ points.)

• •

For line Nikodym sets a lower bound $(1-o(1))q^{n}$ is given in [6] where the implied constant is independent of $n$ but depends on the characteristic of $\mathbb{F}_{q}$.

• •

Improved lower bounds on (line) Kakeya and Nikodym sets in $\mathbb{F}_{q}^{3}$ are given in [10]. In particular it is shown that a construction for Nikodym sets in $\mathbb{F}_{q}^{3}$ of size $\frac{q^{3}}{4}+O(q^{2})$ cannot exist and Nikodym sets behave differently than Kakeya sets where we have such a construction, see our first remark.

• •

Modular conics, in particular hyperbolae, are well-studied objects. For a survey on modular hyperbolae see [12].

• •

The proofs of the lower bounds for the size of finite field Kakeya and Nikodym sets were inspired by ideas from coding theory, see for example [7, Chapter 4] and [14], more precisely from decoding Reed-Muller codes. The crucial idea is that a single missing value of a polynomial (of sufficiently small degree) on a line can be recovered. Similarly one can design decoding algorithms using non-degenerate conics instead of lines, see [14, Lemma 2.6] for parabolae.

• •

The following example shows for ellipses we can neither take $\underline{b}$ nor $\underline{c}$ as a direction to define elliptic Kakeya sets and prove a lower bound of order of magnitude $q^{n}$. We take $n=2$, $q\equiv 3\bmod 4$ and $q\geq 19$. Note that $q\equiv 3\bmod 4$ if and only if $-1$ is a non-square in $\mathbb{F}_{q}^{*}$, that is, for any non-square $g$ in $\mathbb{F}_{q}^{*}$ the element $-g$ is a square in $\mathbb{F}_{q}^{*}$ and let $r\in\mathbb{F}_{q}^{*}$ be a square-root of $-g$, $r^{2}=-g$. Moreover, verify that $\mathbb{F}_{q}^{2}\setminus\{\underline{0}\}=\{(c_{1}r^{-1},c_{2}):(c_{1},c_{2})\in\mathbb{F}_{q}^{2}\setminus\{\underline{0}\}\ \}.$ Then set

[TABLE]

which defines only one ellipse $\mathcal{E}=\{(x,y):y^{2}=gx^{2}+1\}$ with $q+1<\left(\frac{q-1}{4}\right)^{2}$ points by [8, Lemma 6.24] and since $q\geq 19$. Note that $c_{1}^{2}+c_{2}^{2}=0$ with $(c_{1},c_{2})\neq(0,0)$ is not possible since $-1$ is a non-square in $\mathbb{F}_{q}$ for $q\equiv 3\bmod 4$. This example explains why we did not include the case of ellipses into the definition of conical Kakeya sets.

Acknowledgment

The authors are supported by the Austrian Science Fund FWF Project P 30405-N32.

We would like to thank the anonymous referees for their valuable comments, in particular for pointing to [5] and [2].