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func. ineq. over M2(Z):det=1

Source: VTRMC 2014 P6

May 10, 2021

functional equationFunctional inequalityinequalitiesMatriceslinear algebra

Problem Statement

Let

S

denote the set of

2

by

2

matrices with integer entries and determinant

1

, and let

T

denote those matrices of

S

which are congruent to the identity matrix

I\pmod3

(so

\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T

means that

a,b,c,d\in\mathbb Z,ad-bc=1,

and

3

divides

b,c,a-1,d-1

).
(a) Let

f:T\to\mathbb R

be a function such that for every

X,Y\in T

with

Y\ne I

, either

f(XY)>f(X)

or

f(XY^{-1})>f(X)

. Show that given two finite nonempty subsets

A,B

of

T

, there are matrices

a\in A

and

b\in B

such that if

a'\in A

,

b'\in B

and

a'b'=ab

, then

a'=a

and

b'=b

. (b) Show that there is no

f:S\to\mathbb R

such that for every

X,Y\in S

with

Y\ne\pm I

, either

f(XY)>f(X)

or

f(XY^{-1})>f(X)

.

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