282 P. Erdös and C. A. Rogers is te caracteristic function of te set El = U {K +Xi + g}, gea for 1 <i,<n. Suppose 0 < k < N and consider te function r

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1 ACTA ARITHMETICA VII (1962) Covering space wit convex bodies by P. ERDÖS and C. A. ROGERS (Toronto) 1. A few years ago Rogers [1] sowed tat, if K is any convex body in n -dimensional Euclidian space, tere is a covering of te wole space by translates of K wit density less tan nlogn+nloglogn+5n, provided n > 3. However te fact tat te covering density is reasonably small does not imply tat te maximum multiplicity is also small. In te natural covering of space by closed cubes, te density is 1, but eac cube vertex is covered 2 n times. Our object in tis note is to prove tat, provided n is sufficiently large, tere is, for eac convex body K, a covering wit density less tan nlogn=, nloglogn ;-4n, and suc tat no point is covered more tan e {n log n + nloglogn + 4n} times. By dimension teory, some points must be covered n + 1 times. 2. In tis section we take K to be a Lebesgue measurable set wit finite positive measure V. Furter let A be te lattice of all points wit integral coordinates, and suppose all te distinct translates of K by te vectors of A are disjoint. We suppose tat N points x1, x,,..., xn are cosen at random in te cube C of points x wit 0<xl <1, 0<x2 <1,..., 0<xfl <1, and investigate te average density of te set of points covered exactly k times by te system of sets (1) K+xi +g (1 < i < H, g e A). Let e (x) be te caracteristic function of te set K. Ten a=(x) = e(x - xj- g) CEA

2 282 P. Erdös and C. A. Rogers is te caracteristic function of te set El = U {K +Xi + g}, gea for 1 <i,<n. Suppose 0 < k < N and consider te function rk(x ) I (-1)-k k! ( _ k) } =k 1 s=1 were te sum I ' is taken over all selections of distinct integers from 1, 2,..., N. Suppose tat x is a point of space, wic belongs to just r of te sets (1). Ten x belongs to just r of te sets X1, E2,..., EN. Tus, if r < k, we ave rk(x) = 0, and, if r > k, we ave r!{-1) - k r! rk(x) _ k! ( - k)! A! (r - )! =k _ r! k! (r-k)! =1, if r=k, =0, if r>k. Tus Tk(X) is te caracteristic function of te set Ek of points belonging to just k of te sets (1). Since Ek is periodic wit period 1 in eac coordinate, its density is equal to te measure of Ek n C, and so is given by 8(Ek ) = f rk(x)dx. C Te mean value SW (o (Ek)) of tis density, taken over all coices of te points x 1, x 27..., XN in C is But, using Fubini's teorem, N M (S (Ek)) = f... f { r rk(x) dx} dx l..., AN. C C C f... fi f 1 1 Ci8(x)dx} dx 1... dxn C c C 8-1 = f { f o1 (x)dx1 dx a C = ' {f JQ(x-xl -g)dx1 } dx C C ged = f { f o(x-x 1)dx1 } dx a = V. ijx)),

3 N Covering space wit convex bodies 283 Hence, te mean value of te density of E k N reduces to ~l(b(e'k)) - V!(-1) -k N! V GL k!(-k)!!(n-)! =k _ N! Vk 1-VN-k k! (N -R)! ( ) 3. In tis section we take K to be a convex set in n-dimensional space of volume V, and establis te existence of a covering of space by translates of K, wic as bot its density and its maximum multiplicity reasonably small. By a result of Rogers and Separd [2] tere is a lattice X11 of determinant 4"V suc tat te distinct translates of K by te vectors of A1 are disjoint. Tus, after applying a suitable linear transformation to K, we may suppose tat te volume V of K is 4 - " and tat te distinct translates of K, by te vectors of te lattice A of points wit integral coordinates, are disjoint. Now take N to be te integer nearest to and to be te integer nearest to 4'"{nlogn+nloglogn+4n}, e{nlogn+nloglogn.+4n}. If F is te set of points covered by or more of te sets of te system it follows from 2, tat K+xi +g (1 < i < N, g e 11), cjil(b(f))= N! Vk(1-V)N-k '~ k! (N- k)! k= N- H! V(1-V! (-AT - )! ( a )n' - 't i! (N-)! (+t)!(h--t)! (1-V)~. Hence In tis sum te ratio of te (t +1) - st term to te t - t term N--t V Y- V +t+1-1-v < +1 1-V < 1 ' 00 IV? JVl(b(F )) <!( ' (1-V)N - (N- V! (Nt=0 )! V `+1 1-V. _ N! N- (+1)(1-V) -!(H-)!V (1-V) (+1)- (N+1)V ' is

4 28 4 P. Erdös and C. A. Rogers So, using Stirling's formula, log M(b(Fi)) < (N- ) log(i + N )-logp.+ By te coice of V, N and, tis yields Similarly, if +(N-)log(1-V)- - flog (l- N)-log (l- (+1)V) -x-0(1). log x(b (F )) < - nlogn- n log log n - 4n + 0 (1). _ 1 '~ 2nlogn' te mean value of te density of te set Eo of points belonging to no set of te system is so tat Tis (1-2'7)K+x ;-}-g (I< i < N, g e A), 59 (b(e,)) = (1-(1-217)nV)N ; l l log~l(b(eo)) = Nlog 1-4--(1- n 1 lgn!! <-N4-n 1- nl 1ogn) " _ -N4-n 1l- lo1n +0 (1ogn1 2ll )) _ -nlogn-nloglogn-3n+0 (nloglogn) ` log J n Provided n is sufficiently large, we ave log 7rt(b(F )) < -nlogn-nloglogn-nlog8-log2, log7,1(b(eo)) < -nlogn-nloglogn-nlog8-log2. ensures tat `~'l(b(f))+~yt(b(e ))<(8nlogn)" -'7'V. Tus we can coose te points x.., X2)..., xn so tat (2) b(i'')+b(eo) < nnv.

5 We prove tat te system of sets Covering space wit convex bodies 28 5 (3) (1-r7)K+xi+g (1 < i < N, g E A), forms a covering of te wole of space wit te required properties. Let x be any point of space. Consider te system of sets -?7K+x+g (g e A). No two distinct sets of te system ave any common point. So te density of te system is rjnv > S (EO). Hence tere will be a point of a set of te system wic does not belong to EO. Tus, for some i wit 1 < i < N, for some points k1, k2 of K, and for some points g1, g2 of A, we ave Consequently belongs to te set -?1k1+x+g 1 = (1-2,q)k2 +xi+g2. x = (1-2a7)k2+rlk1+xi+(g2- g1) (1 - rj)k+xi+(g2 - g1) of te system (3). Tis sows tat te system (3) covers te wole of space. Now suppose tat a point x of space was covered or more times by te system of sets (3). Ten eac point of te set r1k+x is covered at least times by te sets of te system (1). So F contains te union U {qk+x+g}. gea But tis set as density?7nv. Hence 8(F,,) > j7nv contrary to (2). Tis sows tat no point of space is covered by te system (3) wit multiplicity or more, and completes te proof. References [1] C. A. R o g e r s A note on coverings, Matematika 4 (1957), pp [2] C. A. Rogers and G. C. Separd, Te difference body of a convex body, Arciv der Matematik, 8 (1957), pp , 5. Recu par la Redaction le