Newman, D.J.

Richards, Ian

Elsevier - Publisher Connector 

JOURNAL OF NUMBER THEORY 10, 385-394 (1978) 
Finite Measures Designed for Accuracy 
on Arithmetic Progressions 
D. J. NEWMAN 
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 
AND 
IAN RICHARDS 
Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 
Communicated by E. Hlawka 
Received July 1, 1976 
Anyone familiar with the so-called “sieving process” knows that it consists 
of repeated removals and insertions of various arithmetic progressions, and 
that the sizes of these progressions can only be estimated crudely. Motivated 
by these considerations we set ourselves the problem of producing a proba- 
bility measure on (1, 2, 3 ,..., n} such that “many” arithmetic progressions 
have measure “close to” l/a, where a is their common difference. 
For exampleconsider n = 12 and attach the measure (l/60, l/30,1/15,1/10, 
2/15,3/20,3/20,2/15, l/10, l/15, l/30, l/60). The fact then is that allthe A.P.‘s 
with common difference a < 6 have exactly their “correct” measure l/u. 
For example the progression 5k + 2 which is 2, 7, 12 has measure l/30 $ 
3/20 + l/60 = l/5; also the progression 3k + 1, i.e., 1, 4, 7, 10 has the 
measure l/60 + l/10 + 3/20 + l/15 = l/3, etc. 
Now if we were to require of our measure that all A.P.‘s of difference a < m 
have exactly the correct measure, l/a, then we could only achieve this for 
m < C(n1j2), C a constant (cf. Theorem 3 below). We find, however, that 
we can obtain almost exact agreement well past this point. Indeed we can get 
up to m = nl+ with errors of size e-+*, where 6 lies somewhere between E and 
4E. 
For convenience we write S,,, to represent the A.P. {ak + b, --co < k < a~}; 
more precisely, when considering measures p supported on the finite set 
(1, 2, 3,..., n}, we mean by ,!& the intersection of the A.P. {ak + b) with the 
interval {I,..., n}. 
385 
0022-314X/78/0104-0385$02.00/0 
Copyright 0 1978 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
386 NEWMAN AND RICHARDS 
THEOREM 1. Given n and m there exists a probability measure TV on 
(1, 2, 39.Y n> such that 
as long as 1 < a < m. 
We will also show that to some extent this construction is as good as can be. 
THEOREM 2. No meaSure p on { 1, 2,..., n}, positive or not, can satisfy 
/ p[Sasb] - I/a I < e-C(nln)4 
for all a, b with 1 < a < m if m > 2CW4. 
(Here C = l,OOO,OOO. Note that taking a = 1 gives the total measure of 
(1, z-9 n} as being very close to 1.) 
A case of interest is that of m = rF, 0 < E < $ where we conclude 
roughly that we can achieve I p[SJ - l/a 1 < e+’ and that we cannot 
achieve I &S,,,] - l/a / < e+*‘. The gap between E and 4~ cannot be nar- 
rowed by our present methods, but our guess is that the truth is closer to E, 
i.e., that Theorem 2 is the one that needs improvement. Indeed Theorem 1 
has a rather simple proof while Theorem 2 has a complicated one, and this 
tends to make one believe that Theorem 1 is closer to the true state of affairs. 
If we ask instead that all of the A.P’s considered have exactly the right 
measure, then we obtain the following result. 
THEOREM 3. Given n and m, a necessary and su@cient condition for the 
existence of a nonzero signed measure TV on { 1,2,3,..., n} such that ~L[S,,,] = l/a 
for all a, b with a < m is 
(where 9 denotes Euler’s function). (*I 
Note. Of course this allows us to compute, for any m, the smallest value 
of n which is compatible with it. Thus form = 6, we obtain n = 12 as in the 
example at the beginning of this article. To construct the corresponding 
measure p, we proceed as follows: First let pk denote the measure of the kth 
point. Then 
pk = the coefficient of zk 
in the polynomial cz * g,(z), where 
FINITE MEASURES 387 
0, denotes the ath cyclotomic polynomial, and l/c is the least common 
multiple of all the numbers <m. 
Now it is well known that 
f da) - (3/.rr2) * m2, 
lZ=l 
and thus the asymptotic relationship between the number of points n and the 
“limit of accuracy” m is: 
n - (3/7r2) * m2, 
so 
m - (r/3112) * n1i2. 
The optimal measure described above is unique. As Thomas Vehka has 
observed, it is not always positive. If we wish to have a positive measure, then 
we can obtain it by increasing the ratio n/m” by a constant factor, which for 
large m and n is asymptotic to .rr2/8. 
THEOREM 3a. Given any n and m, there exists a probability measure TV on 
(1, L.., nj such that &9,,,] = l/a for all a, b with a < m provided that 
m < (&1/3)l/~ - 1. 
Unlike Theorem 3, this result is probably not best possible. We suggest the 
problem: For large m and n, does there exist a nontrivial positive measure on 
(1, L..., n} giving an exact equidistribution of mass on congruence classes 
(mod a) for all a < m, and where 
n < (3/rr2) ’ m2 + o(m2)? 
The proof that the measure in Theorem 3 is not always positive is given at 
the end of this article. 
Proofs of Theorems l-3. In our proofs we will exploit the connection 
between the measures of A.P.‘s and the values of the “characteristic function” 
at the roots of unity. Thus calling y(z) = Cy pkzk (where pk denotes the 
measure of the point k) we have the two-way relationship 
(A) If w  is an ath root of unity then 
(W PFL~J = ; 1 w-~T(w) 
0 allatll 
roots Oil 
388 NEWMAN AND RICHARDS 
[(A) is trivial and (B) follows directly from (A) using the fact that Cwa=t w” = 
{a if a I v or 0 if not}.] 
From (A) we obtain 
(I) If I &&J - l/a I d E for all b then I v(w)\ 6 a . E for all dh 
roots of unity w  # 1. 
While (B) gives 
(II) If I y(w)l < E for all dh roots of unity other than I and if v( 1) = 1 
then / p[SQ,J - I/a ( < E. 
Let us fix an m and, for convenience, call W the set of all ath roots of unity 
for all a < m. Theorems 1 to 3 thus result from 
THEOREM 1’. There exists an nth degree polynomial q(z) with nonnegative 
coejkients having q(l) = 1 and such that I q(z)\ < lO~(~/~)(~f~) throughout 
w - {I}. 
THEOREM 2’. If V(Z) = lL,zn + *.a + plz is any nth degree polynomial 
such that ) v(z)] < e- 106(nlm)4 throughout W - (1) and if m > 20n3/* then 
I dl)l < 4. 
For Theorem 3, since we seek precision, it is convenient to replace the 
interval {I, 2 ,..,, n> by (0, l,..,, n - 1). Thus here the number of points n is 
one more than the degree of the corresponding polynomial. 
THEOREM 3’. q(z) = 0 for all z E W - { 1} if and only if the polynomial 
q(z) is divisible by 
(**I 
where !Da denotes the ath cyclotomic polynomial. 
THEOREM 3a’. The polynomial 
h,(z) = n (1 + z + z2 + .*. + z’“-l) 
m/2<aQn 
has positive coeficients, is of degree <(3/8)(m2 - I), and is divisible by the 
polynomial g,Jz) of Theorem 3’. 
[Recall that, since our polynomials have a zeroth term, the number n of 
coefficients is ‘one more than the degree. Thus to deduce Theorems 3 and 3a 
from 3’ and 3a’ requires a little attention to detail. For Theorems 3 and 3’, 
the formulas (*) and (**) match because the product in (**) starts with a = 2. 
FINITE MEASURES 389 
In Theorem 3a, the term “- 1” could be dropped, giving the simpler 
relation m < (8n/3)rj2, by making a careful enumeration of cases (m even/ 
m odd) and by considering the greatest integer in the square root. Since 
Theorem 3a (unlike Theorem 3) is not best possible, there seems to be no 
point in pursuing the matter.] 
Proof of Theorem 1’. We begin with the familiar &function identity 
Cr=‘=_, e-nxr2e2nikt = x-l/2 E:,“=_, e-(n/z)(v+t)a where we will suppose 0 < x < I 
and -+ < t < Q. 
We obtain an approximate equality from this by truncating the left side 
past the terms -42 < k < n/2 while at the same time truncating the right 
side past the single term v = 0 (and later on setting x = 2/mn). Thus we 
obtain 
where 
j E j < 2 c e-a=@ + -.$ v$ e-(~l*)(+1/2)2. 
+%I2 
Now we have 
,<zr2 e-~~k2 < 1 e-nz(n2/4+ia) < e-nX(n2/4) (1 + 1”” e-nrue du) 
i>O 
= pr.(n2/11 
! 
1 1 + - 
2x’ I2 1 
< & e-nx(n2/4) 
while 
jJ e- (n/zNu-l/2l2 < , e-n/4z(1 + e-n/” + e-2”/2 f . ..) 
and so I E / < (3/x1/2)(e-rrz(n’/4) + e-11/42). 
Now we shall think of these series as functions of t (or more properly of 
e2?rit), and let u be the value corresponding to t = 0. We find: 
CT= -n,&<n,2 Psk2 = &T f E’, 
where E’ satisfies the same inequality as E above. 
Let us now define the desired function CJX 
e2ni[n/2]t 
y(e2nit) = -T--- 
vnp$gn,a e--nmk2eanikt. 
390 NEWMAN AND RICHARDS 
For e2nit E W - (1) we have 1 t \ > 11 m and so we may estimate this 9) by 
X-U2~--nl~~e + E ( e-nlxm~ + 3(e-nxCna/4) + piax) 
- 
x-W $ E’ ’ 1 _ 3(pxw/4) + p/42) ’ 
and if we finally choose x = 2/mn we obtain the bound 
as required. 
We turn to Tfieorem 2’: Now the points of W break the unit circle into arcs 
(the so-called Farey arcs); if the arc A has for endpoints a primitive a,th root 
of 1 and a primitive a,th root of 1 then we say it is of type (a1 , uz) and we write 
r(A) = min(a, , a2). We also write /(A) for the length of A. 
Some simple facts in the elementary theory of Farey arcs are that 
(1) If A is of type (a, , a2) then Z(A) = 2n/a,a, . 
(2) If (a 1 , a2) is a type of some A then a, + u2 > m, and a simple 
consequence of these is that 
(3) I(A) < h-/m *r(A). 
Let us fix a numberj and agree to call an arc a major arc if r(A) < j and a 
minor arc if r(A) 3 j E N. We also introduce the set S consisting of all the 
midpoints (on the circle) of pairs of points each from a major arc, and we 
proceed to estimate the measure of this set S. If we fix on two (not necessarily 
distinct) major arcs Al and A,, then the set of midpoints between the points 
from A, and the points from A, consists of two arcs each of length Q(I(A,) + 
)(A,)). Altogether then, we have 
I SI < c W,) + 4-42) = 2 c 44. c 1 
A,.A, major A major A major 
=2x C 44.~ c 1, 
r<j r(A)=r r<j r(A)=r 
and from (3) and the fact that the number of arcs with r(A) = r is 29(r) < 2r, 
this is in turn bounded by 
and so we obtain 
(4) 1 S 1 < (16v/m)j3. 
For convenience at this point we introduce the norm llf(z)ll = ~upl~l,~ 
1 f(z)] (on functions, J which need not necessarily be analytic). 
FINITE MEASURES 391 
We now prove 
LEMMA I. Let P(z) be apolynomiul of degree <rnj/4n such that j P(z)! < 1 
throughout W - ( 1). 
Then, except on the set S, 1 P(z)\ < (2 jl P //)1/2. 
Proof. Fix z $ S and form the trigometric polynomial r(S) = P(<)P(z2/i). 
For each c either 5 or z2/[ lies on a minor arc (or else z, their midpoint, 
would belong to S). Suppose w.1.o.g. that 5 lies on a minor arc, B, and let o 
be the nearer endpoint of B to 5; note that w  f 1, so that \ T(w)\ = j P(o)\ * 
I P(z2b)l < 1 * II p Il. 
Now, by Bernstein’s theorem we have (1 T’ // < mj/4r j( T/j, and so j T(c)\ = 
I T(w) +- Jt T’(s) ds I B 11 P II + mjJ4, II T Ii * 41(B) so that, by (3), 1 T(i3 < 
jl P I/ + 4 II T/l. Varying C gives I( Tll d /I P [j + 4 I/ Til and we obtain 
\I T j/ < 2 (1 P Ii. Finally, setting 5 = z in the definition of T, we obtain 
T(z) = P”(z) which gives us j P”(z)\ < 2 /j P Ij as required. 
LEMMA 2. If P(z) is an nth degree polynomiaI such that ) P(z)\ d 1 on 
/ z / = 1 except for a set, T, of measure OL < 1, then 11 P lj < e2an. (Here the 
measure of the circle is taken to be 2rr.) 
[This is not the best possible result, such being that jj P jj & T,(sec ~~14) 
whether or not LY 9 I ; the proof is essentially contained in [2]. The weak 
version above is all that we require and so we include the simple proof.] 
Proof. By Jensen’s formula we have, for r < 1, 
log / P(eit)[ dt 
1 - 2r cos(8 - t) + r2 
and so 
(l-r2 log I/ P (j 
’ ___ 2?r s 
log I P(e9l dt ( 1 - r2 
T (1 - r)” ‘- yfy * (1 * % 
log I P(reis)l < 1 + r CL 
r” -,i-=-;-2;;log!lP!I+nlog~. 
The maximum, over 8, of [ P(reie)i/r” is the maximum modulus of P(z)/z” on 
/ z ) = r and, by the maximum modulus theorem (applied to the function 
P(z)/zn in a “disk about the point at infinity”), this bounds the maximum 
modulus of P(z)/z” on ( z / = 1, i.e., j/ PI!. 
Thus we obtain 
a l+r 1 
log II P II < 7jp 1 _ r log II P II + n log; . 
392 NEWMAN AND RICHARDS 
Setting r = e-* and observing that 
1 + e-u 1 + e-1 
----<-<e-1 1 - e-a 
this gives log 11 P /I ,< $ log jl P /I + na, the desired result. 
Combining (4) and our two lemmas we easily obtain the following: 
LEMMA 3. Let P(z) be a polynomial of degree <mj/h and suppose that 
16nj3/m < 1. If 1 P(z)1 < I throughout W - (I) then I/ P // < 2e1Q4. 
Proof Lemma 1 gives j P(z)/ < (2 11 P (1)1/2 except on S, which by (4) has 
measure < 16rjs/m d 1. Lemma 2 is therefore applicable to P(z)/(2 11 P ll)V 
and it gives jl P /I/(2 11 P 11)1/2 < e (2~16nis/m)(mj/4n) = &3ip, as required. 
Theorem 2’ can now be read off from this lemma. Simply choose j as the 
first integer above 47r(n/m). Then surely a(z) does have degree n ,< mj/4r 
and we do have 167rj3/m < (I 6n/m)(4n(n/m) + 1)3 < 105(n3/m4) < 1, since 
m > 20n9f4. 
Lemma 3 applied to P(z) = e108(nlm)4 . q(z) then gives e10B(nlm)4 11 cp jj < 
2e16(4s(nlm)+l)’ which easily implies I/ y I[ < + as required. Theorem 2’ is 
proved. 
Proof of Theorems 3’ and 3a’. This is almost trivial. For Theorem 3’: 
We know from our general “characteristic function” argument that a 
polynomial 
gives exact equidistribution of mass on (0, 1 ,..., n - l> for all congruence 
classes (mod a) for any particular u if and only if y(z) vanishes at all ath roots 
of unity other than 1. In other words, we have: 
LEMMA. The sequence of numbers p. , pl ,..., p+l gives the same total mass 
to each congruence class (mod a) on {O, 1 ,..., n - l} if and only if the poly- 
nomial v(z) is divisible by 
1 + 2 + 22 + **. + za-1 
(where Qp, = the dth cyclotomic polynomial). 
Note. The above lemma means that the measure pO, pL1 ,..., pn-l is a 
convolution product of some measure {vk) with the measure (1, 1, 1,. .., 
1, 0, 0, O,...} consisting of a “l’s” followed by an infinite string of “0’s.” 
FINITE MEASURES 393 
The lemma can be proved purely combinatorially, and then using it, 
Theorems 3 and 3a can be proved algebraically without invoking Fourier 
analysis. We doubt, however, whether the same thing is true for Theorems 1 
and 2. 
Theorems 3’ and 3a’ are immediate consequences of the lemma. We 
conclude with two “complements” to Theorem 3 which were observed by 
Thomas Vehka. 
The first is that, if we take the measure (J.LJ determined by the polynomial 
(so that g,(z) = ,u,, + plz + -a- + ps-lzn-l), theff each value pk is an 
integer, and the total mass of the interval (0, l,..., n - l} is equal to the least 
common multiple of all the numbers <m. 
To see this, just note that the total mass = g&l), and that each Q,(l) = 
either p or 1, depending on whether a is a prime power pk or not. 
[Since each pk: is an integer, the “equipartition” (mod a) for all a < m 
trivially implies that the total mass would have to be a common multiple 
of all the numbers a < m. What is remarkable is that it is always the least 
common multiple.] 
The second observation is that the measures determined by the polynomials 
g,(z) are not necessarily positive. Thus let pk.m = the coefficient of zk in 
g&z) = the mass at the integer k corresponding to g,(z). Then there are 
infinitely many values of m for which pI,m is negative. For, remembering that 
and using the standard formula for the cyclotomic polynomial Qi, , we obtain 
711 
P I,,,I = - Uz p(u) (where p(u) is the Miibius function). 
* 
Finally, it is well known1 that the series C ~(a) oscillates between + co and 
-co; in other words, the numbers pLl,m. oscillate in sign, assuming both 
positive and negative values of arbitrarily large magnitude.2 
1 The oscillation of X p(a) follows, via a standard Tauberian theorem of analytic number 
theory, from the fact that the function’l/&) has no singularities on the real axis. 
* The first value of m for which pI,% < 0 is m = 95. We do not know whether there is 
any smaller value of m for which some other coefficient pr,nL, k f  I, is negative. 
394 NEWMAN AND RICHARDS 
REFERENCES 
1. H. HALBERSTAM AND H.-E. RICHERT, “Sieve Methods,” Academic Press, London, 1974. 
2. REMET, Sur une propikt6 des Polynomes de Tebebychev, Comm. Inst. Sci. Kharkov 13 
(1936), 93-95. 


Finite measures designed for accuracy on arithmetic progressions 

file:///data/remote/core/dit/data/elsevier/pdf/852/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDIyMzE0eDc4OTAwMTQ4.pdf

Finite measures designed for accuracy on arithmetic progressions

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Elsevier - Publisher Connector