Good Functions and Multivariable Polynomials
Saarik Kalia and Michael Zanger-Tishler
Third Annual MIT PRIMES Conference
May 18, 2013
Good Functions
I
The Oppenheim Conjecture, which concerns representations
of real numbers by real quadratic forms, was formulated in
1929 by Alexander Oppenheim and proved by Grigory
Margulis (who won the Fields Medal in 1978) in 1986 using
new methods invented by Margulis.
Good Functions
I
The Oppenheim Conjecture, which concerns representations
of real numbers by real quadratic forms, was formulated in
1929 by Alexander Oppenheim and proved by Grigory
Margulis (who won the Fields Medal in 1978) in 1986 using
new methods invented by Margulis.
I
Later, the Sprindžuk-Baker Conjecture was proved by
Margulis and Kleinbock, our project advisor, using a
quantitative version of Margulis’s method, and a key
ingredient was the use of (C , α)-good functions.
Definition
I
For C , α > 0, a function f : Rn → R is (C , α)-good if for
every ball B ⊂ Rn and > 0,
α
f ,
λn (B ) ≤ C
λn (B).
kf kB
Definition
I
For C , α > 0, a function f : Rn → R is (C , α)-good if for
every ball B ⊂ Rn and > 0,
α
f ,
λn (B ) ≤ C
λn (B).
kf kB
I
kf kB := supx∈B |f (x)|.
I
B f , := {x ∈ B : |f (x)| < }.
I
If kf kB = 0, we let
I
λn denotes the Lebesgue measure in Rn .
1
0
= ∞.
Example
α
λn (B)
kf kB
√
f (x) = x 2 , C = 2 2, α = 21
λn (B f , ) ≤ C
Example
α
λn (B)
kf kB
√
f (x) = x 2 , C = 2 2, α = 21
B = [−1, 1] =⇒ λ1 (B) = 2, kf kB = 1
λn (B f , ) ≤ C
Example
α
λn (B)
kf kB
√
f (x) = x 2 , C = 2 2, α = 21
B = [−1, 1] =⇒ λ1 (B) = 2, kf kB = 1
λn (B f , ) ≤ C
= 0.5
= 0.3
= 0.1
Example
α
λn (B)
kf kB
√
f (x) = x 2 , C = 2 2, α = 21
B = [−1, 1] =⇒ λ1 (B) = 2, kf kB = 1
λn (B f , ) ≤ C
= 0.5
λ1
(B f , )
=
√
2
λ1
= 0.3 q
(B f , )
=
6
5
= 0.1 q
λ1 (B f , ) =
2
5
Example
α
λn (B)
kf kB
√
f (x) = x 2 , C = 2 2, α = 21
B = [−1, 1] =⇒ λ1 (B) = 2, kf kB = 1
λn (B f , ) ≤ C
= 0.5
√
λ1
(B f , )
√
2≤2 2
=
√
0.5
1
2
1
2
λ1
·2
q
6
5
= 0.3 q
(B f , )
√
≤2 2
=
0.3
1
= 0.1 q
6
5
1
2
·2
q
λ1 (B f , ) = 25
√ 0.1 1
2
2
·2
5 ≤2 2 1
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
I
f is (C , α)-good if and only if g (x) = |f (x)| is also
(C , α)-good.
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
I
f is (C , α)-good if and only if g (x) = |f (x)| is also
(C , α)-good.
I
If f is (C , α)-good, then so is g (x) = kf (x) for all k ∈ R.
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
I
f is (C , α)-good if and only if g (x) = |f (x)| is also
(C , α)-good.
I
If f is (C , α)-good, then so is g (x) = kf (x) for all k ∈ R.
I
If fi , i ∈ I are all (C , α)-good, then so is g (x) = supi∈I fi (x).
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
I
f is (C , α)-good if and only if g (x) = |f (x)| is also
(C , α)-good.
I
If f is (C , α)-good, then so is g (x) = kf (x) for all k ∈ R.
I
If fi , i ∈ I are all (C , α)-good, then so is g (x) = supi∈I fi (x).
I
If f is (C , α)-good, then so is g (x) = f (x + a) for all a ∈ Rn .
Basic Properties
I
If f is (C , α)-good, it is also (C 0 , α)-good for all C 0 > C and
(C , α0 )-good for all α0 < α.
I
f is (C , α)-good if and only if g (x) = |f (x)| is also
(C , α)-good.
I
If f is (C , α)-good, then so is g (x) = kf (x) for all k ∈ R.
I
If fi , i ∈ I are all (C , α)-good, then so is g (x) = supi∈I fi (x).
I
If f is (C , α)-good, then so is g (x) = f (x + a) for all a ∈ Rn .
I
If f is (C , α)-good, then so is g (x) = f (kx) for all k ∈ R.
Single-Variable Polynomials
Kleinbock and Margulis proved:
Theorem
All polynomial functions f : R → R of degree k are
1
(2k(k + 1) k , k1 )-good.
Single-Variable Polynomials
Kleinbock and Margulis proved:
Theorem
All polynomial functions f : R → R of degree k are
1
(2k(k + 1) k , k1 )-good.
Proof.
I
Choose x1 , · · · , xk+1 ∈ B f , such that |xi − xj | ≥
all i, j ≤ k + 1.
λ1 (B f , )
2k
for
Single-Variable Polynomials
Kleinbock and Margulis proved:
Theorem
All polynomial functions f : R → R of degree k are
1
(2k(k + 1) k , k1 )-good.
Proof.
I
Choose x1 , · · · , xk+1 ∈ B f , such that |xi − xj | ≥
all i, j ≤ k + 1.
I
By Lagrange interpolation:
k+1
k+1
X
Y x − xj
f (xi )
f (x) =
xi − xj
i=1
j=1,j6=i
λ1 (B f , )
2k
for
Single-Variable Polynomials
Kleinbock and Margulis proved:
Theorem
All polynomial functions f : R → R of degree k are
1
(2k(k + 1) k , k1 )-good.
Proof.
I
Choose x1 , · · · , xk+1 ∈ B f , such that |xi − xj | ≥
all i, j ≤ k + 1.
I
By Lagrange interpolation:
k+1
k+1
X
Y x − xj
f (xi )
f (x) =
xi − xj
i=1
j=1,j6=i
=⇒ kf kB ≤ (k + 1)
λ1 (B)
λ1 (B f , )
2k
!k
λ1 (B f , )
2k
for
Single-Variable Polynomials
Kleinbock and Margulis proved:
Theorem
All polynomial functions f : R → R of degree k are
1
(2k(k + 1) k , k1 )-good.
Proof.
I
Choose x1 , · · · , xk+1 ∈ B f , such that |xi − xj | ≥
all i, j ≤ k + 1.
I
By Lagrange interpolation:
k+1
k+1
X
Y x − xj
f (xi )
f (x) =
xi − xj
i=1
λ1 (B f , )
2k
j=1,j6=i
=⇒ kf kB ≤ (k + 1)
λ1 (B)
!k
λ1 (B f , )
2k
1
=⇒ λ1 (B f , ) ≤ 2k(k + 1) k
kf kB
1
k
λ1 (B).
for
Project Goals
I
Can we find optimal C ’s and α’s for these or other families of
functions?
Project Goals
I
Can we find optimal C ’s and α’s for these or other families of
functions?
I
A group of Brandeis undergraduates showed that C =
optimal for single-variable polynomials.
2k
√
k
k!
is
Project Goals
I
Can we find optimal C ’s and α’s for these or other families of
functions?
I
I
A group of Brandeis undergraduates showed that C =
optimal for single-variable polynomials.
Can we explore multivariable polynomials?
2k
√
k
k!
is
Project Goals
I
Can we find optimal C ’s and α’s for these or other families of
functions?
I
I
A group of Brandeis undergraduates showed that C =
optimal for single-variable polynomials.
2k
√
k
k!
Can we explore multivariable polynomials?
I
Kleinbock and Margulis showed that n-variable functions of
1
degree k are (C , nk
)-good for some C .
is
Project Goals
I
Can we find optimal C ’s and α’s for these or other families of
functions?
I
I
A group of Brandeis undergraduates showed that C =
optimal for single-variable polynomials.
2k
√
k
k!
is
Can we explore multivariable polynomials?
I
Kleinbock and Margulis showed that n-variable functions of
1
degree k are (C , nk
)-good for some C .
Conjecture
All polynomial functions f : Rn → R of degree k are (C , k1 )-good
for some C .
Multivariable Linear Polynomials
Theorem
All linear polynomial functions f : Rn → R are
1
4Vn−1
Vn , 1
Here Vn stands for the volume of the unit ball in Rn , i.e.
V0 = 1, V1 = 2, V2 = π, V3 = 4π
,···.
3
-good.1
Multivariable Linear Polynomials
Theorem
All linear polynomial functions f : Rn → R are
4Vn−1
Vn , 1
-good.1
Proof.
I
1
Let r be the radius of B and c be the perpendicular distance
from its center to the hyperplane f (x) = 0.
Here Vn stands for the volume of the unit ball in Rn , i.e.
V0 = 1, V1 = 2, V2 = π, V3 = 4π
,···.
3
Multivariable Linear Polynomials
Theorem
All linear polynomial functions f : Rn → R are
4Vn−1
Vn , 1
-good.1
Proof.
I
I
1
Let r be the radius of B and c be the perpendicular distance
from its center to the hyperplane f (x) = 0.
qP
P
n
2
Let f (x) = ni=1 ai xi and let d =
i=1 ai .
Here Vn stands for the volume of the unit ball in Rn , i.e.
V0 = 1, V1 = 2, V2 = π, V3 = 4π
,···.
3
Multivariable Linear Polynomials
Theorem
All linear polynomial functions f : Rn → R are
4Vn−1
Vn , 1
-good.1
Proof.
I
I
I
1
Let r be the radius of B and c be the perpendicular distance
from its center to the hyperplane f (x) = 0.
qP
P
n
2
Let f (x) = ni=1 ai xi and let d =
i=1 ai .
Then, kf kB = d(c + r ) and the distance between the
hyperplanes f (x) = and f (x) = − is 2
d .
Here Vn stands for the volume of the unit ball in Rn , i.e.
V0 = 1, V1 = 2, V2 = π, V3 = 4π
,···.
3
Multivariable Linear Polynomials
Theorem
All linear polynomial functions f : Rn → R are
4Vn−1
Vn , 1
-good.1
Proof.
I
I
Let r be the radius of B and c be the perpendicular distance
from its center to the hyperplane f (x) = 0.
qP
P
n
2
Let f (x) = ni=1 ai xi and let d =
i=1 ai .
I
Then, kf kB = d(c + r ) and the distance between the
hyperplanes f (x) = and f (x) = − is 2
d .
I
We have four cases:
1
Here Vn stands for the volume of the unit ball in Rn , i.e.
V0 = 1, V1 = 2, V2 = π, V3 = 4π
,···.
3
Multivariable Linear Polynomials
d
≥r +c
r < c, d < c − r
r ≥ c, d ≤ r + c
r < c, c − r ≤
d
<c +r
Multivariable Linear Polynomials
I
Case 1: B f , = B =⇒ Trivial
I
Case 2: B f , can be bounded by hypercylinder of height 2
d
α
4Vn
n−1
f
,
λn (B)
and base Vn−1 r
=⇒ λn (B ) ≤ Vn−1 kf kB
I
Case 3: B f , = ∅ =⇒ Trivial
I
Case 4: B f , can be bounded by hypercylinder of height
+ r − c and base
α
n
λn (B)
Vn−1 r n−1 =⇒ λn (B f , ) ≤ V2V
kf kB
n−1
Multivariable Quadratic Polynomials
Theorem
P
f (x) =
n
2
i=1 xi
is
4Vn−1 1
Vn , 2
-good.
Multivariable Quadratic Polynomials
Theorem
P
f (x) =
I
n
2
i=1 xi
is
4Vn−1 1
Vn , 2
-good.
This proof is similar to that of the linear polynomials, except
we intersect two balls rather than a ball and the region
between two hyperplanes.
Multivariable Quadratic Polynomials
Theorem
P
f (x) =
n
2
i=1 xi
is
4Vn−1 1
Vn , 2
-good.
I
This proof is similar to that of the linear polynomials, except
we intersect two balls rather than a ball and the region
between two hyperplanes.
I
Interestingly, this case gives better C ’s than the optimal C for
the entire family of quadratic polynomials, i.e. where the
√
optimal C for single-variable quadratic polynomials is 2 2 the
optimal C for specifically this function (f (x) = x 2 ) is 2.
General Case
Theorem
All polynomial
functions f : R2 → R of degree k are
8k 1
√
, k -good.
k
k!
General Case
Theorem
All polynomial
functions f : R2 → R of degree k are
8k 1
√
, k -good.
k
k!
Proof.
I
Draw chords through the supremum of f on B, yielding
f , )
is
single-variable polynomials. For each chord l, λλ1 (l∩B
(l∩B)
1
1 f , )
k
2k
bounded by √
.
so we can bound λλ2 (B
k
kf kB
2 (B)
k!
General Case
Theorem
All polynomial
functions f : R2 → R of degree k are
8k 1
√
, k -good.
k
k!
Proof.
I
Draw chords through the supremum of f on B, yielding
f , )
is
single-variable polynomials. For each chord l, λλ1 (l∩B
(l∩B)
1
1 f , )
k
2k
bounded by √
.
so we can bound λλ2 (B
k
kf kB
2 (B)
k!
I
The problem reduces to how best to maximize such a region
f , )
when λλ1 (l∩B
is a fixed p.
1 (l∩B)
General Case
Theorem
All polynomial
functions f : R2 → R of degree k are
8k 1
√
, k -good.
k
k!
Proof.
I
Draw chords through the supremum of f on B, yielding
f , )
is
single-variable polynomials. For each chord l, λλ1 (l∩B
(l∩B)
1
1 f , )
k
2k
bounded by √
.
so we can bound λλ2 (B
k
kf kB
2 (B)
k!
I
The problem reduces to how best to maximize such a region
f , )
when λλ1 (l∩B
is a fixed p.
1 (l∩B)
I
Letting c be the distance from the center of the circle to the
supremum, we want to spread the points of the region as far
away from the supremum as possible.
General Case
c=0
c=
1
2
c=
3
4
c=
9
10
c=1
General Case
Lemma
Let R be a subset of circle S such that for every chord l of S
through some point P, λλ11(l∩R)
(l∩S) is at most p. Then
λ2 (R)
λ2 (S)
< 4p − 2p 2 .
General Case
Lemma
Let R be a subset of circle S such that for every chord l of S
through some point P, λλ11(l∩R)
(l∩S) is at most p. Then
λ2 (R)
λ2 (S)
< 4p − 2p 2 .
1
2 2 !
k
k
2k
2k
λ2 (B f , ) < 4 √
−2 √
λ2 (B)
k
k
kf kB
k! kf kB
k!
1
k
8k
< √
λ2 (B).
k
k! kf kB
Future Research
I
We would like to prove that the value of k1 is the optimal
value for k degree multivariable polynomials.
Future Research
I
We would like to prove that the value of k1 is the optimal
value for k degree multivariable polynomials.
I
We would like to optimize our values for C . The estimations
we used to get our values of C are clearly not optimal and we
hope to lower our value of C .
Related Works
BPS S. Bacon, J. Pardo and G. Sturm, (C , α)-good functions,
Brandeis University course project 2011.
DM S.G. Dani and G.A Margulis, Limit distributions of orbits of
unipotent flows and values of quadratic forms, Adv. in Soviet
Math. 16 (1993), 91-137.
KM D. Kleinbock and G.A Margulis, Flows on homogeneous
spaces and Diophantine approximation on manifolds, Ann.
Math. 148 (1998), 339-360.
KT D. Kleinbock and G. Tomanov, Flows on S-arithmetic
homogeneous spaces and applications to metric Diophantine
approximation, Comm. Math. Helv. 82 (2007), 519- 581.
Acknowledgments
I
We would like to thank a number of people for enabling our
research:
Acknowledgments
I
We would like to thank a number of people for enabling our
research:
I
Professor Dmitry Kleinbock for proposing our project and
providing us guidance on our projects the last two years
Acknowledgments
I
We would like to thank a number of people for enabling our
research:
I
Professor Dmitry Kleinbock for proposing our project and
providing us guidance on our projects the last two years
I
Our mentor Tue Ly, for providing us tips for the problem and
dedicating countless hours to making this project a success
Acknowledgments
I
We would like to thank a number of people for enabling our
research:
I
Professor Dmitry Kleinbock for proposing our project and
providing us guidance on our projects the last two years
I
Our mentor Tue Ly, for providing us tips for the problem and
dedicating countless hours to making this project a success
I
The MIT PRIMES program for enabling the two of us to work
together the last two years
Acknowledgments
I
We would like to thank a number of people for enabling our
research:
I
Professor Dmitry Kleinbock for proposing our project and
providing us guidance on our projects the last two years
I
Our mentor Tue Ly, for providing us tips for the problem and
dedicating countless hours to making this project a success
I
The MIT PRIMES program for enabling the two of us to work
together the last two years
I
And last but not least, our parents for providing us with cars
and rides to get back and forth from our meetings.