ON THE DIAPHONY OF ONE CLASS OF ONE-DIMENSIONAL SEQUENCES

In the present paper, we consider a problem of distribution of sequences in the interval [0,1), the so-called ’Pr-sequences’ We obtain the best possible order O(N-(logN) /’2) for the diaphony of such Pr-sequences For the symmetric sequences obtained by symmetrization of Psequences, we get also the best possible order O(N-l(lo9N) 1/.2) of the quadratic discrepancy

From (1.1) and (1.2) becomes clearly that the best possible order of diaphony and quadratic discrepancy of every sequence a in E is O(N-(logN) 1/2 ).
2. A SEQUENCE OF r-ADIC RATIONAL TYPE.
We signify No N U {0), with N the set of natural integers.A sequence of r-adic rational type (or RP-sequence) is a sequence ((i)),:0, which is generated by the guiding matrix (v,j) in the following way: If in the r-adic number system i-8mem_ 1 then in the r-adic number system where for j 1, 2, , m Wj eV VfVf "V, ( e terms and is the operation ofthe digit-by-digit addition modulo r of elements of Z {0, 1, , r 1}. A RP-sequence (o(i)),=0, which is generated by the guiding matrix (v,l) can be also construct:l by following the three mentioned below rules () (0) O.

es+ terrr8
Obviously the operation has commutative and associative property.
We shall prove that the two definitions of the PR-sequenes are equivalent.
Let us suppose that the first definition is valid for RP-sequence.
Reversely, let the second deflation for PR-sequence is valid d is given positive integer.Then there ests uquely positive integer s that r S < r+1.We shall prove deflation by induction on s.Ifs 0, then I < r and (i) iv()'(0) 0, ivy.
We make inductive supposition that for some s N and eve integer i, r -1 S < r deflation holds.
Deflation holds for eve positive integer .
The proof ofthe lena is obvious.
For eve integer a Z we define the oy integer, weh is a lution of the equation a + O(mod r).
A r-adie element inte is inte t, [(-1)/,j/), in weh I S j S r , for y imeger m.
Let N r.We shl 1 the net X (z0, z, ., zu_) be a net of e P (or P-e), if eve r-adie element inte l,s, hating lenh 1/N ntn one point ofthe net X.
A r-adie section of the sequence X (z),0 is a set of tes z,, th numbers i, mtisng the inequities kr < ( + 1)r , for eve integers k and , such that k 0, 1, The sequence (z,)io is eled a sequence of te P (or P-suenee) if eve r-adie section is a P-net.
OM 2.1.Let in the iding mat (v,S) eve v. 1 d for j > eve v,s 0, i.e., Then the corresponding RP-sequence is Pr-sequence PROOF.We choose arbitrary r-adic section of the RP-sequence (o(i)),0, the length of which is r'.We write the numbers i, belonging to this section in the r-adic number system: where ck are fixed and ek are arbitrary r-adic numbers We choose now an arbitrary r-adic interval l, with length r-'-In the r-adic system this interval is determined by the inequality O a a2 a, < x < O a a2 "am +0, where al, , a, are r-adic numbers We shall prove, that for every choice of the numbers ck and ak among the numbers i, in the form (2.4) there exists exactly one i, for which o(i) E l.
In this system the unknowns el, e2, ", ern are successively so determined that it has only one solution.
The theorem is proved.
In the following lemma we shall show some property of Pr-sequences.LEMMA 2.2.Let N r" where u E No.For every guiding matrix (v,o) in which v,,, 1 and v,, 0 for j > s(s 1, 2, and for the RP-sequence (o(i)),__0, which is product of (v,,j) we have {o(i): 0 < < r"} {fiN: 0 _< j < N} (2.6) PROOF.We shall make the proof by induction on u.If u 0 and u 1, then we make directly examination.
Consecutively we solve the left over equations and get uniquely integer number l'= (l'ul'), such that 0 < I" < r-1.
Finally, we establish a bijection between the sets from the two sides ofthe equation (2.16).
The lemma is proved.
3. AN ESTIMATION FROM ABOVE FOR THE DIAPHONY OF Pr-SEQUENCES.
THEOREM 3.1.Let in the guiding matrix (v,) every v,,, 1 and for j > s every v,, 0 and let a (o(i)),__ o be the P-sequence which is produced by the (v,).Then for every positive integer N we have Fg(cr) <_ c(r)g-l(log((r 1)g + 1)) 1/2, where the constant c(r) is given by c(r) 7r((r 1)/3 log r) 1/. ( The proof of this theorem is based on a non-trivial estimate for the trigonometric sum of an arbitrary P, sequence. 3.1.AN ESTIMATION OF THE TRIGONOMETRIC SlIM OF ARBITRARY Pr-SEQUENCE.Let X ,),,=0 is arbitrary sequence in interval E.A trigonometric sum, SN(X:h), of the sequence X, where h is an integer is the quantity SN(X;h) N-1 -=0 exp (27rihx).
The proof of lemma is obvious.
Then for every integer h we have SN (X; h) < E,=la S(,_l)r,r-(X; h) The proof of lemma is based ofLemma 3.1 and is done by induction on a.
Let in the guiding matrix (v,) every v, 1 and for j > s every vs, 0 and cr (o(n)),__ 0 be the Pr-sequence which is product of (v,).
Then for every integer h we have SN (a; h) < Ej:0 aj r, (h) PROOF.Let N > 1 be an integer with r-adic representation of a type (3.2).
We shall prove that for every integer h and for every sequence X in interval E we have the estimation [SN (g; h) < :=0 Em=ll Sire-l/r, (S; h)[ (3.3)where we have the supposition that when a 0, the inside sum is 0.
Let h be an integer.For eve N 1 ests an integer n, such that N < r.We shl prove the lena by the induction on n Ifn 1, then the estimation (3.3) is tribal.
We suppose, that (3.2) is te for eve integer N, 1 N < r , where n is some integer.
Let now N such that r N<r +.By herewe have, that in (3.2) a=0 forj>n Let N P + Q where P ar and Q a.
Let now j, 0 j n be arbitra fixed number and consider that 1 m a.If m 1, then by Lemma 2.2 for the trigonometric sum S (a; h) we have s(; h)= , (h) (3.4) (a; h), by Lena 2.4, we have Let now 2 m a. Then for the trigonometric sum S(m_) (ih((m ))) E:0 zp (ih(k)).
Let (vs,) is an arbitrary guiding matrix, such that on principal diagonal there stand ones, and over him zeros and a (,#,(n)),__ 0 is Pr-sequence, which is bred by the matrix (vs.).
are integer numbers.