CONTRIBUTIONS TO THE ERDÔS' CONJECTURE ON ARITHMETIC PROGRESSIONS

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DEMONSTRATIO MATHEMATICA
Vol. XLII No 2 2009
Roman Wituia, Damian Slota
CONTRIBUTIONS TO THE ERDÔS' CONJECTURE ON
ARITHMETIC PROGRESSIONS
Abstract. The most famous unsolved is Erdos' conjecture that every set A C N such
that a~1 =00 contains arbitrarily long arithmetic progressions. In this paper a new
a eA
aspects of this one are presented.
1. The first contribution
The most famous unsolved is Erdos' conjecture that every set A C N such
that a~1 = 00 contains arbitrarily long arithmetic progressions. Erdos
aeA
has estimated the solution as worth 3000$ (see [2,3]). In this paper we do
not present a solution of this great Erdos' problem. We will touch only a few
new aspects of this one and clarify it a little further.
In view of that, in [4] W. Sierpinski proved the following two interesting
lemmas:
LEMMA 1. Let {«„} be any sequence of positive integers. Then there exists
a sequence {vn}^=1 C N such that limn^oo un/vn = 0 and the set
contains no infinite arithmetic progression.
LEMMA 2. be an increasing sequence of positive integers. Then
there exists a sequence {vn}^Li of positive integers increasing faster than the
sequence {un}^=1 (in the sense: linin^oo urJvn = 0), which contains finite
arithmetic progressions of any length.
The above Lemmas became now completed with the following relevant
result.
LEMMA 3. Let be an increasing sequence of positive integers such
that = 00. Then there exists an increasing sequence of positive integers
2000 Mathematics Subject Classification: 11B25.
Key words and phrases: Erdos' problem, arithmetic progression.

266 R. Witula, D. Slota
{vn}^=i satisfying the following conditions: E = oo; lim ^ = 0 and the
sequence {vn}^Li contains arithmetic progressions of any length.
Proof. Let {kn}^=1 and be two auxiliary sequences of positive
integers that meet the following conditions:
(a) kn+i - kn > ln > k\\
(b) ln+i > ll\
(c) E - ^
r—l+kn
(d) Ukn+1 < ln (Ukn+1 i In) for i = 0, 1, . . .
,
ln - 1.
As far as condition (d) is concerned, it should be observed that for every
i G {1,...,
fcn_|_i},
we have:
Ukn+1-i < ukn+1 ~i.
Now, the sequence shall be determined in the following manner:
Vkn+1-i = ll {ukn+1 -iln) for i = 0,1, ...,/„ - 1,
and
Vkn+i— i
— In
ukjl+1
~i
for i — ln, ln "I-1,
• •
•,
kn-\-1
kn 1,
and ne N.
Furthermore, we set Vi = i for i = 1, 2,..., k\. It should be noted that
1) in view of (d) and by the definition of vn we have:
<for i = 0,1,... ,kn+i - kn - 1;
Ukn+l~i / 1
vkn+i~i In
2) the subsequence {vkn+l-n+i, vkn+1-n+2,
• •
-,vkjl+1} is an arithmetic pro-
gression with the difference term equal to —Z^;
kn+l In fcn+1 In
3) E è = E ¿rr > 1 for each n € N, which in view of (c) follows;
r=l+fcn r=l+k„
4) condition (d) also guarantees that vkji+1-in+1 > vkn+1-in, i.e., the sequence
{vn
'•
kn + 1 < n < kn+1} is increasing;
5) in view of (a) and (b) the sequence {vn}^=1 is increasing.
Consequently, it follows from 1) that lim un/vn = 0; whereas from 3)
n—>00
00
that E IT = Next, the condition 2) means that the sequence
n=1
contains arithmetical progressions of any length. •

Contributions to the Erdôs' conjecture 267
2. The second contribution
Let 5 denote the family of all nonincreasing sequences {ari}^L1 of positive
reals such that
lim an = 0 and > an = oo.
n—>oo '
Let {pn}r^=i denote the increasing sequence of all prime numbers. In [7] the
following two results are proven:
THEOREM 4. For a given sequence {an} € 5 there exists a sequence {bn} £ 5
such that ^min{an,i)n} < oo
iff
lim inf n an = 0.
THEOREM 5. Let {an}, {an} e 6. //liminf nan = 0 and liminf ana"1 = 0
then there exists a sequence {6n} G 5 such that
min{o:n, bn} = +oo and min{an, bn} < oo.
Moreover, we need also the following two known facts:
THEOREM 6. ([5]) We have
lim = 1.
n—HX)
n log n
THEOREM 7. If{xn}^=1 is nondecreasing sequence of positive integers then
— = oo <*=>• ^^ = oo.
xn n xn
Using all above theorems we get the following two corollaries, which are
our second contribution to Erdos' problem.
THEOREM 8. By Theorems 4, 6 and 7 there exists an increasing sequence
of positive integers such that
El = oo and ^^ min| —, — 1 < oo.
n
L
"n Pn
J
We note, that by Ex. 1 of [7] this sequence {bn}^=1 can be obtained effectively!
THEOREM 9. Let a € (0,1]. By Theorems 5, 6 and 7 there exists an
increasing sequence {cnj^j of positive integers such that
y^mini—, — 1 = oo and y^minl—,— — 1 < oo.
^ [cn pn) ^ [cn pn (LOGN)A J
REMARK 10. The last two theorems are very attractive in comparison with
T. Tao and B. Green beautiful result (see [1,6]) that there exist arbitrarily
long arithmetic progressions of prime numbers.

268 R. Witula, D. Slota
References
[1] B. Green, T. Tao, Restriction theory of the Selberg sieve, with applications, J. Théor.
Nombres Bordeaux 18 (2006), 147-182.
[2] T. tuczak, Z. Palka, Paul Erdôs, Wiadom. Mat. 33 (1997), 99-109 (in Polish).
[3] A. Sarkozy, Paul Erdôs (1913-1996), Acta Arith. 81 (1997), 301-317.
[4] W. Sierpinski, Remarques sur les progressions arithmétiques, Colloq. Math. 3 (1954),
44-49.
[5] W. Sierpinski, Elementary Theory of Numbers, PWN and North-Holland, Warsaw,
1987.
[6] T. Tao, Arithmetic progressions and the primes, Collect. Math. Vol. Extra (2006),
37-88.
[7] R. Witula, On the convergence of series having the form min{an, bn}, Nieuw. Arch.
Wisk. 10 (4), no. 1-2 (1992), 1-6.
INSTITUTE OF MATHEMATICS
SILESIAN UNIVERSITY OF TECHNOLOGY
Kaszubska 23
GLIWICE 44-100, POLAND
E-mail: d.slota@polsl.pl
Received October 15, 2007; revised version December 22, 2008.