Uniform Distribution and Waring s Problem with digital restrictions

Transcription

1 Uniform Distribution and Waring s Problem with digital restrictions Manfred G. Madritsch Department for Analysis and Computational Number Theory Graz University of Technology madritsch@tugraz.at Colloque international sur la répartition uniforme 22 janvier 2008 joint work with Jörg M. Thuswaldner, University of Leoben. Supported by the Austrian Research Fund (FWF), Project S / 19

2 Outline Digital restrictions Number systems over F q [X ] Number systems over F q [X, Y ]/p(x, Y )F q [X, Y ] Summary 2 / 19

3 Number systems over Z Let q 2 be an integer. Then every positive integer n can be represented in the form n = l 0 d l q l. We call a function f strictly q-additive if it acts on the q-ary digits of a number. Thus f (n) = l 0 f (d l ). 3 / 19

4 Number systems over Z Let q 2 be an integer. Then every positive integer n can be represented in the form n = l 0 d l q l. We call a function f strictly q-additive if it acts on the q-ary digits of a number. Thus f (n) = l 0 f (d l ). 3 / 19

5 The digitally restricted set Let f 1,..., f r be strictly q-additive functions and j 1,..., j r and m 1,..., m r be positive integers. Then we define the set S := {n N : f 1 (n) j 1 (mod m 1 ),..., f r (n) j r (mod m r )}. 4 / 19

6 Waring s problem and uniform distribution We look for an asymptotic formula for the number of solutions of n = x k x k s, x 1,..., x s S. We order the elements of S by the sequence (s i ) i 0. Is for h a polynomial with at least one irrational coefficient the sequence (h(s i )) i 0 uniformly distributed modulo 1? 5 / 19

7 Waring s problem and uniform distribution We look for an asymptotic formula for the number of solutions of n = x k x k s, x 1,..., x s S. We order the elements of S by the sequence (s i ) i 0. Is for h a polynomial with at least one irrational coefficient the sequence (h(s i )) i 0 uniformly distributed modulo 1? 5 / 19

8 Exponential sum The main rôle in the proof is played by the following exponential sum ( ) r r i e h(n) + f i (n) m i n N where e(x) := exp(2πi x). In order to estimate the exponential sum one has to consider correlations of the form h 1 H 1 h k H k e n N i=1 ( ri m i k (f i (n); h 1,..., h k ) ) 2. 6 / 19

9 Exponential sum The main rôle in the proof is played by the following exponential sum ( ) r r i e h(n) + f i (n) m i n N where e(x) := exp(2πi x). In order to estimate the exponential sum one has to consider correlations of the form h 1 H 1 h k H k e n N i=1 ( ri m i k (f i (n); h 1,..., h k ) ) 2. 6 / 19

10 Waring s Problem with digital restrictions Theorem Thuswaldner,Tichy The equation n = x k x k s, x 1,..., x s S, has always a solution for sufficiently large n. 7 / 19

11 Outline Digital restrictions Number systems over F q [X ] Number systems over F q [X, Y ]/p(x, Y )F q [X, Y ] Summary 8 / 19

12 Q-ary digital expansion Q-ary digital expansion: Fix Q F q [X ], then for A F q [X ] A = i 0 D i Q i (deg D i < deg Q). Strongly Q-additive: A function f : F q [X ] F q [X ] is called strongly Q-additive if f (A) := f (D i ). i 0 9 / 19

13 Q-ary digital expansion Q-ary digital expansion: Fix Q F q [X ], then for A F q [X ] A = i 0 D i Q i (deg D i < deg Q). Strongly Q-additive: A function f : F q [X ] F q [X ] is called strongly Q-additive if f (A) := f (D i ). i 0 9 / 19

14 The general setting Fix Q i -additive functions f i (1 i r) and consider the set S := {A F q [X ] : f 1 (A) J 1 (M 1 ),..., f r (A) J r (M r )}. By (S l ) l 0 we denote a sequence through all elements of S such that m n deg S m S n. 10 / 19

15 The general setting Fix Q i -additive functions f i (1 i r) and consider the set S := {A F q [X ] : f 1 (A) J 1 (M 1 ),..., f r (A) J r (M r )}. By (S l ) l 0 we denote a sequence through all elements of S such that m n deg S m S n. 10 / 19

16 The exponential sum In this number field the exponential sum looks similar to that in Z. Thus we have to consider ( ) n 1 r r i E h(z l ) + f i (Z l ) m i l=0 where (Z l ) l 0 is a sequence of all elements of F q [X ] such that i=1 for all m, n N. m n deg Z m deg Z n 11 / 19

17 Corresponding problem of Waring Theorem M,Thuswaldner The equation N = P k P k s, ( Pi S, deg P i < ) deg N k always has a solution provided N has sufficiently large degree. Theorem M,Thuswaldner Let h be a polynomial of degree 0 < k < p = char F q. Then the sequence h(s i ) is uniformly distributed in K if and only if at least one coefficient of h(y ) h(0) is irrational. 12 / 19

18 Corresponding problem of Waring Theorem M,Thuswaldner The equation N = P k P k s, ( Pi S, deg P i < ) deg N k always has a solution provided N has sufficiently large degree. Theorem M,Thuswaldner Let h be a polynomial of degree 0 < k < p = char F q. Then the sequence h(s i ) is uniformly distributed in K if and only if at least one coefficient of h(y ) h(0) is irrational. 12 / 19

19 Outline Digital restrictions Number systems over F q [X ] Number systems over F q [X, Y ]/p(x, Y )F q [X, Y ] Summary 13 / 19

20 Y -ary digital expansion Y -ary digital expansion: If p(x, Y ) is monic in X and Y then every A F q [X, Y ] has an unique and finite expansion by A = i 0 D i Y i (D i F q [X ], deg D i < deg X p). Strongly Q-additive: A function f : F q [X, Y ] F q [X, Y ] is called strongly Y -additive if it acts on the D i only. E.g. the sum of digits function, which is defined by s Y (A) := i 0 D i. 14 / 19

21 Y -ary digital expansion Y -ary digital expansion: If p(x, Y ) is monic in X and Y then every A F q [X, Y ] has an unique and finite expansion by A = i 0 D i Y i (D i F q [X ], deg D i < deg X p). Strongly Q-additive: A function f : F q [X, Y ] F q [X, Y ] is called strongly Y -additive if it acts on the D i only. E.g. the sum of digits function, which is defined by s Y (A) := i 0 D i. 14 / 19

22 The general setting We fix only one Y -additive f and consider the set S := {A F q [X, Y ] : f (A) J (M)} 15 / 19

23 The exponential sum The exponential sum in this area looks like the following E (h(a) + RM ) f (A), A B(n) where B is the set of integers in F q (X, Y )/p(x, Y )F q (X, Y ) over F q [X ]. 16 / 19

24 Waring s Problem Theorem M,Thuswaldner There always exists a solution for N = P k P k s (P i (B(m) S)) provided that N is sufficiently large. 17 / 19

25 Outline Digital restrictions Number systems over F q [X ] Number systems over F q [X, Y ]/p(x, Y )F q [X, Y ] Summary 18 / 19

26 Summary q-additive functions Uniform Distribution Waring s Problem Z F q[x ] F q[x, Y ] Kim Drmota & Gutenbrunner Madritsch & Thuswaldner Thuswaldner Madritsch & Madritsch & & Tichy Thuswaldner Thuswaldner 19 / 19

27 T. Beck, H. Brunotte, K. Scheicher, and Jörg M. Thuswaldner, Number systems and tilings over Laurent series, submitted (2007). Mireille Car, Le problème de Waring pour l anneau des polynômes sur un corps fini, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A141 A144., Waring s problem in function fields, Proc. London Math. Soc. (3) 68 (1994), no. 1, L. Carlitz, Diophantine approximation in fields of characteristic p, Trans. Amer. Math. Soc. 72 (1952), A. Dijksma, Uniform distribution of polynomials over GF{q, x} in GF[q, x]. I, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), / 19

28 , Uniform distribution of polynomials over GF{q, x} in GF[q, x]. II, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 (1970), M. Drmota and G. Gutenbrunner, The joint distribution of Q-additive functions on polynomials over finite fields, J. Théor. Nombres Bordeaux 17 (2005), no. 1, Dong-Hyun Kim, On the joint distribution of q-additive functions in residue classes, J. Number Theory 74 (1999), no. 2, R. M. Kubota, Waring s problem for F q [x], Dissertationes Math. (Rozprawy Mat.) 117 (1974), 60. M. G. Madritsch and J. M. Thuswaldner, Waring s Problem in Function Fields with digital restrictions, manuscript. 19 / 19

29 , Weyl Sums in F q [x] with digital restrictions, submitted. K. Scheicher and J. M. Thuswaldner, Digit systems in polynomial rings over finite fields, Finite Fields Appl. 9 (2003), no. 3, J. M. Thuswaldner and R. F. Tichy, Waring s problem with digital restrictions, Israel J. Math. 149 (2005), , Probability in mathematics. W. A. Webb, Waring s problem in GF[q, x], Acta Arith. 22 (1973), / 19