Bounds on sets with few distances modulo a prime in metric spaces of strength t

Bounds on sets with few distances modulo a prime in metric

spaces of strength t

Citation for published version (APA):

Blokhuis, A., & Singhi, N. M. (1981). Bounds on sets with few distances modulo a prime in metric spaces of strength t. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8116). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981 Document Version:

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1981-16 December 1981

Bounds on sets with few distances modulo a prime in metric spaces of strength t

by

A. B10khuis - N.M. Singhi

"

Technological University Department of Mathematics &

Computing Science PO Box 513, Eindhoven The Netherlands

Bound on sets with few distances

modulo a prime in metric spaces of strength t

by

A. Blokhuis - N.M. Singhi

r~

O. Abstract

.~

In this paper we prove an extension and slight generalization of a theorem of FRANKL and WILSON. [3J

Theorem: Let (X,d) be a metric space of strenght t, and B c X. For a prime p and integer s let

(i) d (a,b) 2 E IN for all a,b

(ii) d (a,b) ; 0 (mod p) 2 for

(iii)

#

{d2 (a, b)(mod p)

I

b E B}

s

Then card B::; L dim Harm(s). i==O

t. Preliminaries

E B

all a,b E B

::; s ::; !t for all a E B.

All statements, propositions and definitions in this section are quoated from [tJ (chapter 9).

Let (X,d) be a metric space of finite diameter

18.

Call c _xy whence 0 ::; c ::;

o.

Let w denote a finite measure on X, with

xy w(X) =

J

dw x w < 00 2 d (x,y),

then w induces a measure W on X x X. Let S be the set {c I x, Y E X} •

.e

2

-Then

w

induces a finite measure II on S. For A € S

ll(A)

=

l

@{{x,y}

I

c € A} •

w xy

The set of polynomials l;x;x ; ... 2 can be orthonormalized w.r.t. the inner-product

<f,g> =

J

f(a)g(a)dll(a)

S

to give a set of polynomials {q.}oo 1 i=O

f

q.(a)q.(a)dll(a) 0 ••

1 J 1J

S

such that

where q. is a polynomial in one variable of degree i.

1

Definition: A metric space (X,d,w) has strength t if

J

ci cj dw(x) = f .. (c b)

ax bx 1J a

v ..

1,]

X

where f .. denotes a polynomial of degree S min{i,j} • 1J

[Comment: A metric space of strength t for all t is called a Delsarte space. Delsarte spaces of finite degree (i.e. where S is a finite set) are essentially the Q-polynomial schemes. Other examples are the compact symmetric spaces of rank 1, the real sphere, a real, complex or quaternion projective space, or the Cayley projective plane. A t-design, or spherical t-design, considered

as a metric space by itself, is a metric space of strength t (cf. [IJ , page 66)J.

Proposition:

J

q.(c )q.(c_b )dw(x)

=

q.(O)q.(c b)o .. if i + j S t .

1 ax J x 1 1 a 1J

.e

3

-Definition: Harm(i) is the space of functions on X generated by the set

{x + q.(c )

I

a E X} . 1 ax

Remark:

By

the proposition above we have Harm(O) i ••• l Harm(rt / Z') with respect to the nondegenerate innerproduct:

<f,g>

I

f(x)g(x)dw(x)

X

Proposition: For i ~ r!t' dim Harm(i) = q. (x) 2 W 1

Remark: The actual values of dim Harm(i) for the projective spaces can be found in HOGGAR [4J.

Notation: H(s) := Harm(O) EII ••• EII Harm(s) •

2. Some lemma's

Lemma J: Let s ~

!to

For all X.E X

3x

E H(s) s.t. Vf E H(s): <i,f> = f(x).

Proof: H(s) is finite dimensional, hence isomorphic to its dual. Further-more the innerproduct is nondegenerate.

Lemma 2: Let M be a nonempty finite set of real numbers. Let 7lM be the set of all 7l -linear combinations of elements from M.. Then 7lM c p71M for some prime p implies M = {a} •

4

-~: This is a consequence of Krull's theorem [5J, page 10. We'll give a proof for the sake of completeness.

~M is a finite dimensional vector space over ~. Write the elements of M as vectors over some fixed basis of this vectorspace. For ~ E ~M let p(~) := the minimal exponent of p in all coordinates of m (where

p(Q) :'"

+ 00).

Now obviously p(~ + ~) ~ min{p(~),p(~)}. Therefore:

min mE?lM hence M :: {O} •

3. Proof of the theorem

'"" min mEM p(~)

=

min p(~) '" ~EpM I + min p(~) mEM

We will show that B (where'" is as in Lemma 1) is an independent subset of H(s). Now suppose this is not the case. We then have a dependency relation

A

I

~b

=

0 •

bEB

s For a E B define F (u)

a IT (a. -~ u) and fa(x)

i=1 {cab (mod p)

I

b E B}. This yields Since Fa is a polynomial Now:

L

~ < b, fa > '" 0 • bEB <b,f > _ 0 (mod p) a if

<a,f >

=

ITa.

i

a

(mod p)

a ~

b f: a

So we get ma E p?lM. where M

=

{~

I

bE B}.

=

F (c ) here _{a ax} {al, .. ·,a_s}

=

of degree s we have f E H{s).

.e

5

-since a was arbitrary we get llM € P llM but from Lermna 2 this implies

~ 0 Vb€B.

This finishes the proof of the theorem.

4. Final remarks

The theorem (and it's proof, with some minor adjustments) is still valid

i f we replace ]N and II by a unique factorization domain D c :R; and <Q in the

proof of Lermna 2 by <Q QII D •

5. References:

[IJ A. Neumaier; Combinatorial configurations in terms of distances, THE memorandum 81-09.

[2J A. Neumaier; Distances, Graphs & Designs,

Europ. J. of Combinatorics (1980) I, 163-174.

[3J P. Frankl & R.M. Wilson; Intersection theorems with geometric conse-quences - to appear in the Europ. J. of Comb.

[4J S.G. Hoggar; t-designs in proj. spaces - to appear in the Europ. J. of Comb. [5J M. Nagata; Local Rings. Interscience tracts in pure and applied