R 0C Z N IK I POLSKIEGO TOW ARZYSTW A MATEMATYCZNEGO Séria I: PEA C E MATEMATYCZNE X V (1971)
A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)
M. K
a l e c k iOn an inequality in the general additive theory of numbers
Let A and В be two increasing sequences of integers
Щ — 0 , cq , 0*2, . . . ^ •••
The increasing sequence formed by the sums 6 q+&?- {%
is denoted by C. Moreover, let us denote the number of terms of A , В and C which do not exceed n (exclusive of 0 ) by A(n), B(n) and C(n), respectively.
Finally,
inf к
A {к)
к and inf
к
в (к) к
are denoted by a and ft. If G(n) < n, then according to the theorem of Mann, G(n) + (a + /?)w. The inequality C(n) + A(n)-\-(in for niO — from which this theorem would easily follow — is not generally true as was shown by Mann (1) by the following example:
A 0 , 1 , 2 , 6 , 7, 8 , 12 , ... -1(11) = 5
В 0, 1, 2 , 6 , 7, 8 , 12 , ... 11 *0 = 1 1 - | = 4 . 4 C 0, 1, 2, 3, 4, 6 , 7, 8 , 9, 10, 12, ... 0(11) = 9 < 5 + 4.4 There arises a problem of finding an inequality for
7 1 4G of the type G {n) + A(n)-\-f{fi)n- We prove below this inequality for /(/3) = /3/(1 + /?).
The proof has some common elements with that of the theorem of Besicovitch (2): G(n) + A{n)-\-(t'n for П
4С, where /S' = in f--- B(k)
к fc + 1 T
h e o r e m. I f
7 1 4G, then
C{n) + A { n ) + Y + J n -
(1) Henry B. M ann, Addition theorems, New York, London, Sydney 1965, p. 22.
(2) Ibidem, p. 20.
134 M. K a l e c k i
Proof. Since the theorem is obvious for = 0 we can confine the proof to the case 1 <гБ.
Let us consider the segments of the sequence C in the interval ( 0 , n) which contain at least one number aeA. Let us number the segments.
The first segment 1 will start from aQ = 0 . The number of segments will be denoted by m. Let us now denote the least a of the segment s by as, the greatest a of this segment by a's' and the greatest c of the segment by c8.
From l e В it follows directly and thus
( 1 ) C(n) > A (n) -f- m .
Next c ' + l 4G and consequently cs+ l ~ b i A . It follows that the numbers c = c '-fl — b satisfying the condition as < c s+ l - b ^ c'g do not belong to A. The number of such numbers is B(cs—a8). We have ( 2 ) C(as+1- 1 ) - C ( a s- 1 )
=
( A(<Vn —1)
— A { d B— l ) ) + B { c 8 — d s )+
( B ( d s + l — a 8 — l ) — B ( c 8 — d s ))= A (ds+l- l ) — A (ds—l) + B{ds+1 — ds- l )
> A { a 8 + 1 — l ) — A ( d 8 — l ) + p { d 8+1 — a 8) — p .
For the least segment we obtain analogously, taking into considera
tion njC,
(3) C ( n ) - C ( d m- l ) ^ A ( n ) - A ( d m- l ) + P ( n - d m+ l ) - j 3 . It follows now from ( 2 ) and (3)
(4) C(n) ^ A(n)-\-(in — pm
and multiplying (1) by /3 and adding it to (4),
( 6 ) G{ n ) ^ A( n ) + Y + J n -
For the sequence quoted at the beginning of the paper we obtain
2 5
