## On some dynamical systems in finite fields and residue rings

Venue: | Discr. and Cont.Dynam.Syst.,Ser.A |

Citations: | 2 - 2 self |

### BibTeX

@ARTICLE{Shparlinski_onsome,

author = {Igor E. Shparlinski},

title = {On some dynamical systems in finite fields and residue rings},

journal = {Discr. and Cont.Dynam.Syst.,Ser.A},

year = {},

pages = {2007--11098}

}

### OpenURL

### Abstract

We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning. 1

### Citations

819 |
Random number generation and quasi-Monte Carlo methods
- Niederreiter
- 1992
(Show Context)
Citation Context ...length, distribution of element and other properties, of the orbits of such dynamical systems has a long and successful history, which dates back to early works of N. M. Korobov [46], H. Niederreiter =-=[59, 60]-=-, A. G. Postnikov [66] and many other researchers. Here we show that some classical results immediately imply some of these conjectures. We also show that several other conjectures are not correct as ... |

561 |
Finite Fields
- Lidl, Niederreiter
- 1983
(Show Context)
Citation Context ...njectures are not correct as they are stated in [1, 2, 3, 4, 5, 6] and need some adjustments. 1For a prime p and a positive integer n, we denote by IFp n the finite field of pn elements (we refer to =-=[52]-=- for the background information on finite fields). We fix a primitive root ϑ of IFp n and recall that we also have IFp n ∼ = IFp[ϑ]. In particular, IFp n can be considered as an n-dimensional vector s... |

171 |
Sieve methods
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ... that πg(x) ∼ A(g) x log x . Let π(x; k, a) denote the number of primes p ≤ x with p ≡ a (mod k). We need the following relaxed version of the Brun–Titchmarsh theorem, see Theorem 3.7 in Chapter 3 of =-=[39]-=-. Lemma 3.2. For any integers k, a ≥ 1 with 1 ≤ k < x, the bound ( ) x π(x; k, a) = O ϕ(k) log(3x/k) holds. Let P be the set of prime numbers. The following estimate can be derived via partial summati... |

134 |
Quasi-Monte Carlo Methods and Pseudo-Random Numbers
- Niederreiter
- 1978
(Show Context)
Citation Context ...length, distribution of element and other properties, of the orbits of such dynamical systems has a long and successful history, which dates back to early works of N. M. Korobov [46], H. Niederreiter =-=[59, 60]-=-, A. G. Postnikov [66] and many other researchers. Here we show that some classical results immediately imply some of these conjectures. We also show that several other conjectures are not correct as ... |

116 |
Automatic sequences. Theory, applications, generalizations
- Allouche, Shallit
- 2003
(Show Context)
Citation Context ... j−1 . (1) j=0 Clearly, assuming that IFp is represented by the elements of the set {0, 1, . . ., p − 1}, one can view the points 1 p am, m = 1, . . .,M, (2) as M points of an n-dimensional unit cube =-=[0, 1]-=- n . For M = p n − 1 these points form a regular cubic lattice (with only one missing point (0, . . ., 0)). It has also been conjectured by V. I. Arnold [5] that in fact even the first M < p n − 1 pow... |

95 |
On the volume of tubes
- Weyl
- 1939
(Show Context)
Citation Context ...ined for ε > 0 and such that limε→0 b(ε) = 0. Following [49, 64], we define the class Mb of domains Ω ⊆ [0, 1] n for which vol Ω + ε ≤ b(ε) and vol Ω− ε ≤ b(ε). As special case of a result of H. Weyl =-=[69]-=- implies that for domains Ω with a piecewise smooth boundary, one can take b(ε) = O(ε). Lemma 2.3. For any domain Ω ⊆ [0, 1] n with a piecewise smooth boundary vol Ω ± ε = O (ε) 8A relation between D... |

89 |
Estimate for the number of sums and products and for exponential sums in fields of prime order, submitted to
- Bourgain, Glibichuk, et al.
(Show Context)
Citation Context ...ult in several directions. In fact, using the results of J. Bourgain and M.-C. Chang [13], which in turn generalize several recently emerged results of J. Bourgain, A. A. Glibichuk and S. V. Konyagin =-=[14, 15]-=-, one can also study the distribution in intervals of the set (2) for extremely small values of M. For example, see [16] for more details and a version of the bound (4) which is nontrivial provided th... |

73 |
On Artin's conjecture
- Hooley
- 1967
(Show Context)
Citation Context ...= 1 L L∑ ℓ=1 gcd(g,ℓ)=1 Tg(L) ∼ c(g) L log L tg(ℓ) for some constant c(g) > 0 depending only on g (we note that in [5] it is made explicit only for g = 2). We show that the classical result of Hooley =-=[42]-=- on Artin’s conjecture, implies, under the Extended Riemann Hypothesis, that the conjecture (5) is wrong and in fact Tg(L) ≥ L log L exp ( C(g)(log log log L) 3/2) . for some constant C(g) > 0 dependi... |

63 |
Discrepancies and Applications
- Drmota, Tichy, et al.
- 1997
(Show Context)
Citation Context ...ry ⇓ ergodic properties of the corresponding dynamical system 5The link between exponential sums and the distribution in aligned boxes is provided by the Koksma–Szüsz inequality, see Theorem 1.21 of =-=[19]-=-. The link between the distribution in aligned boxes and arbitrary regions is given by the results of H. Niederreiter and J. M. Wills [64] and their more recent refinement of M. Laczkovich [49]. Surpr... |

59 |
Character sums with exponential functions and their applications
- Konyagin, Shparlinski
- 1999
(Show Context)
Citation Context ...Ω) = Mvol Ω + o(M) (3) provided that M ∼ µp n for some fixed µ > 0 (and p → ∞). We start with an observation that using classical bounds of incomplete exponential sums with exponential functions, see =-=[45, 46, 52]-=-, and some standard tools from the theory of uniform distribution, see Section 2.4, one can derive the following improved version of the conjecture (3): Nϑ(M, Ω) = Mvol Ω + O ( M 1−1/n p 1/2n (log p) ... |

48 |
Pointwise ergodic theorems for arithmetic sets
- Bourgain
- 1989
(Show Context)
Citation Context ...sums from [9, 34, 36], one can obtain nontrivial results for even smaller intervals, which however holds only for almost all primes p (rather than for all p). Furthermore, motivated by the results of =-=[11, 32]-=-, we consider the distribution of vectors am where instead of an initial segment [1, M], m runs through the values of a polynomial. Unfortunately, we are not able to treat arbitrary polynomials with i... |

30 | The distribution of integers with a divisor in a given interval
- Ford
(Show Context)
Citation Context ...uences is considered in [37, 54]. It is well known that if an integer g > 1 is fixed then for any function ε(x) with ε(x) → 0 as x → ∞, for almost all primes p the bound tg(p) ≥ p 1/2+ε(p) holds, see =-=[23, 26, 44, 65]-=- for various improvements of this result. For almost all integers ℓ, similar type bounds are given in [47]. It is clear that when g varies, tg(ℓ) runs through divisors of λ(ℓ). In fact through all the... |

29 |
Carmichael's lambda function
- Erdos, Schmutz, et al.
- 1991
(Show Context)
Citation Context ... (6) 4For the upper bound it is probably natural to assume that ( L∑ ) 1 Tg(L) = o λ(ℓ) . (7) L Note the sum on the right hand side of (7) has been estimated by P. Erdős, C. Pomerance and E. Schmutz =-=[24]-=-. Finally, we give a guide to the literature concerning results and methods which can probably be of great use for the theory of algebraic dynamical systems over finite fields and rings. It is very we... |

25 |
Mordell’s exponential sum estimate revisited
- Bourgain
(Show Context)
Citation Context ... a divisor of k, see [10, 53, 68] and the references therein. Certainly similar questions about d(ϕ(ℓ)) are very natural and interesting. 4 Repeated squaring and other nonlinear transformations Using =-=[7, 12, 27, 28, 29, 31]-=- one can also easily derive various uniformity of distribution results for the vectors aem where e ≥ 2 is a fixed integer. Alternatively, these results can be interpreted as results about orbits of re... |

25 |
asymptotische Verhalten von Summen über multiplikativen
- Wirsing, Das
- 1961
(Show Context)
Citation Context ...age value of d(k) behaves like D(K) = 1 K K∑ d(k) ∼ k=1 3K 2 log K is wrong too. Since d(k) is a multiplicative function, so is d(k)/k, which also satisfies the conditions of the Wirsing theorem, see =-=[70]-=-. Thus one can easily show that in fact K D(K) ∼ κ (log K) 1/2 19for some absolute constant κ = 0.4067 . . ., see [10] for this and some other results on the properties of the average divisor, includ... |

25 |
Ergodic Theory of Numbers
- Dajani, Kraaikamp
- 2002
(Show Context)
Citation Context ...nued fractions, various number systems, the 3x + 1 transformation and other number-theoretic constructions. A wealth of old and recently emerging results and references can be found in the monographs =-=[1, 20, 29, 30, 63, 52, 72, 89, 91, 95]-=-. However, we would like to use this paper as an opportunity to attract more attention of the dynamical system community to a great variety of already existing number theoretic results and techniques ... |

24 |
On the distribution of digits in periodic fractions
- Korobov
- 1972
(Show Context)
Citation Context ...that the study of the length, distribution of element and other properties, of the orbits of such dynamical systems has a long and successful history, which dates back to early works of N. M. Korobov =-=[46]-=-, H. Niederreiter [59, 60], A. G. Postnikov [66] and many other researchers. Here we show that some classical results immediately imply some of these conjectures. We also show that several other conje... |

22 |
An introduction to the theory of numbers (Oxford Univ
- Hardy, Wright
- 1979
(Show Context)
Citation Context ... = O M ∑ Dϑ(f; M) 1/n ) ϑ∈Θ ⎛ = O ⎝M ⎛ ( ϕ(p n − 1) ϑ∈Θ = O ⎝Mϕ(p n − 1) 1−1/n n−1 ∑ ϑ∈Θ Dϑ(f; M) ) 1/n ⎞ ⎠ ( ) ⎞ 1/n ∑ Dϑ(f; M) ⎠ . Since k/ϕ(k) = O(loglog k) for every integer k, see Theorem 328 of =-=[40]-=-, from (12) we derive the desired estimate. ⊓⊔ For example, we see from Theorem 2.6 that for every fixed ε and M ≥ p n/2+ε , for almost all primitive roots of ϑ ∈ IFp n, we have Nϑ(f; M, Ω) ∼ Mvol Ω f... |

20 |
On the normal behavior of the iterates of some arithmetic functions, Analytic number theory
- Erdos, Granville, et al.
- 1990
(Show Context)
Citation Context ...( ) x π(x; k, a) = O ϕ(k) log(3x/k) holds. Let P be the set of prime numbers. The following estimate can be derived via partial summation from Lemma 3.2, see, for example, the proof of Theorem 3.4 in =-=[22]-=-. 13Lemma 3.3. For any integer f ≥ 1 the bound holds. ∑ p∈P, f 2 ≤p≤x p≡1 (mod f) 1 p log(3x/p) Proof. Let h = ⌊2 log f⌋, H = ⌈log x⌉. Then ∑ p∈P, f 2 ≤p≤x p≡1 (mod f) ≪ 1 p log(3x/p) ≤ H∑ j=h H∑ j=h... |

19 | Period of the power generator and small values of Carmichael’s function
- Friedlander, Pomerance, et al.
(Show Context)
Citation Context ...ed values of e) although the bounds obtained within this approach are less explicit. These results are complemented by the estimates on the orbit lengths of such transformations which are obtained in =-=[30, 58]-=- and which show that these orbit lengths tend to be large (and close to their largest possible values). The distributional properties of dynamical systems generated by general non-linear transformatio... |

18 |
On the order of a (mod p
- Erdos, Murty
- 1999
(Show Context)
Citation Context ...uences is considered in [37, 54]. It is well known that if an integer g > 1 is fixed then for any function ε(x) with ε(x) → 0 as x → ∞, for almost all primes p the bound tg(p) ≥ p 1/2+ε(p) holds, see =-=[23, 26, 44, 65]-=- for various improvements of this result. For almost all integers ℓ, similar type bounds are given in [47]. It is clear that when g varies, tg(ℓ) runs through divisors of λ(ℓ). In fact through all the... |

18 | On the distribution of matrix elements for the quantum cat map
- Kurlberg, Rudnick
- 2005
(Show Context)
Citation Context ...ons of our results and directions for further research. Although exponential sums have been used for this purpose, see the work of M. Degli Esposti and S. Isola [18] and of P. Kurlberg and Z. Rudnick =-=[48]-=-, their full potential seems to be not fully used in the dynamical system theory. We would like to stress that all such applications follow the same pattern: ℓ=1 bounds of exponential sums ⇓ distribut... |

17 | The distribution of totients
- Ford
- 1998
(Show Context)
Citation Context ...y on g. Furthermore, we believe that in fact Tg(L) grows even faster. It is possible that the method of proof of Theorems 1 and 2 in [8], which in turn is an extension of the method of [57] (see also =-=[25]-=-), together with the result of Hooley [42], can be used to derive that, under the Extended Riemann Hypothesis, Tg(L) ≥ (5) L log L exp ( (log log log L) 2+o(1)) . (6) 4For the upper bound it is proba... |

17 |
t l, A survey of quadratic and inversive congruential pseudorandom numbers
- -Herrmann, Herrmann, et al.
- 1998
(Show Context)
Citation Context ...l systems generated by general non-linear transformations x ↦→ f(x) where f is a rational function, over a finite field or a residue ring, have been extensively studied in the literature as well, see =-=[23, 24, 25, 78, 79, 80, 81, 82, 83, 84, 85, 86]-=- and the references therein. As in the case of repeated powering all these results indicate that if the orbit is long enough then its elements are uniformly distributed. On the other hand, these resul... |

14 |
Timofeev, `Divisors of shifted primes
- Indlekofer, M
(Show Context)
Citation Context ...uences is considered in [37, 54]. It is well known that if an integer g > 1 is fixed then for any function ε(x) with ε(x) → 0 as x → ∞, for almost all primes p the bound tg(p) ≥ p 1/2+ε(p) holds, see =-=[23, 26, 44, 65]-=- for various improvements of this result. For almost all integers ℓ, similar type bounds are given in [47]. It is clear that when g varies, tg(ℓ) runs through divisors of λ(ℓ). In fact through all the... |

12 | On the distribution of the power generator
- Friedlander, Shparlinski
(Show Context)
Citation Context ... a divisor of k, see [10, 53, 68] and the references therein. Certainly similar questions about d(ϕ(ℓ)) are very natural and interesting. 4 Repeated squaring and other nonlinear transformations Using =-=[7, 12, 27, 28, 29, 31]-=- one can also easily derive various uniformity of distribution results for the vectors aem where e ≥ 2 is a fixed integer. Alternatively, these results can be interpreted as results about orbits of re... |

12 |
A mean ergodic theorem for (1/N) ∑N n=1 f(T nx)g(T n2x). Convergence in ergodic theory and probability 92
- Weiss
- 1993
(Show Context)
Citation Context ...sums from [9, 34, 36], one can obtain nontrivial results for even smaller intervals, which however holds only for almost all primes p (rather than for all p). Furthermore, motivated by the results of =-=[11, 32]-=-, we consider the distribution of vectors am where instead of an initial segment [1, M], m runs through the values of a polynomial. Unfortunately, we are not able to treat arbitrary polynomials with i... |

11 |
Estimates on exponential sums related to the Diffie-Hellman distributions, Geom
- Bourgain
(Show Context)
Citation Context ...ult in several directions. In fact, using the results of J. Bourgain and M.-C. Chang [15], which in turn generalize several recently emerged results of J. Bourgain, A. A. Glibichuk and S. V. Konyagin =-=[13, 14, 16, 17]-=-, one can also study the distribution in intervals of the set (2) for extremely small values of M. For example, see [18] for more details and a version of the bound (4) which is nontrivial provided th... |

10 |
On the order of finitely generated subgroups of Q ∗ (mod p) and divisors of p − 1
- Pappalardi
- 1996
(Show Context)
Citation Context |

8 |
Distribution of closed orbits for linear automorphisms of tori, Nonlinearity 8
- Esposti, Isola
- 1995
(Show Context)
Citation Context ...s. We however indicate possible extensions of our results and directions for further research. Although exponential sums have been used for this purpose, see the work of M. Degli Esposti and S. Isola =-=[18]-=- and of P. Kurlberg and Z. Rudnick [48], their full potential seems to be not fully used in the dynamical system theory. We would like to stress that all such applications follow the same pattern: ℓ=1... |

8 | On the period of the linear congruential and power generators - Kurlberg, Pomerance |

8 |
On the number of distinct values of Euler’s φ-function
- Maier, Pomerance
- 1988
(Show Context)
Citation Context ...0 depending only on g. Furthermore, we believe that in fact Tg(L) grows even faster. It is possible that the method of proof of Theorems 1 and 2 in [8], which in turn is an extension of the method of =-=[57]-=- (see also [25]), together with the result of Hooley [42], can be used to derive that, under the Extended Riemann Hypothesis, Tg(L) ≥ (5) L log L exp ( (log log log L) 2+o(1)) . (6) 4For the upper bo... |

7 | The iterated Carmichael λ-function and the number of cycles of the power generator
- Martin, Pomerance
- 2005
(Show Context)
Citation Context ...ed values of e) although the bounds obtained within this approach are less explicit. These results are complemented by the estimates on the orbit lengths of such transformations which are obtained in =-=[30, 58]-=- and which show that these orbit lengths tend to be large (and close to their largest possible values). The distributional properties of dynamical systems generated by general non-linear transformatio... |

6 |
A Gauss sum estimate in arbitrary finite fields
- Bourgain, Chang
- 2006
(Show Context)
Citation Context ...M/p1/2 (log p) n+1 → ∞. In fact, using some results of H. Niederreiter [59, 60] one can easily extend the above result in several directions. In fact, using the results of J. Bourgain and M.-C. Chang =-=[13]-=-, which in turn generalize several recently emerged results of J. Bourgain, A. A. Glibichuk and S. V. Konyagin [14, 15], one can also study the distribution in intervals of the set (2) for extremely s... |

6 |
Polynomial congruences over incomplete residue systems modulo k
- Chalk
- 1989
(Show Context)
Citation Context ...ame order on the largest distance between the points (2) and on the largest radius of a ball inside of [α, β] n which does not contain any points (2). Moreover, using some standard modifications, see =-=[17]-=-, one can drop the logarithmic factor from these bounds. 3 Dynamical systems in residue rings 3.1 Preliminaries Given an integer g ≥ 2 with gcd(g, ℓ), V. I. Arnold [1, 3, 5, 6] suggests to consider th... |

6 |
Some doubly exponential sums over Zm
- Friedlander, Konyagin, et al.
(Show Context)
Citation Context ... a divisor of k, see [10, 53, 68] and the references therein. Certainly similar questions about d(ϕ(ℓ)) are very natural and interesting. 4 Repeated squaring and other nonlinear transformations Using =-=[7, 12, 27, 28, 29, 31]-=- one can also easily derive various uniformity of distribution results for the vectors aem where e ≥ 2 is a fixed integer. Alternatively, these results can be interpreted as results about orbits of re... |

6 |
Certain Exponential Sums and Random Walks on Elliptic Curves
- Lange, Shparlinski
(Show Context)
Citation Context ... matrix has been considered in [2]. Using the results and methods of [38], one can prove various uniformity of distribution properties of orbits of such dynamical systems. Accordingly, the results of =-=[7, 41, 50]-=- can be used to obtain similar statements for analogues of the above dynamical systems on elliptic curves over finite fields. Acknowledgements The author wishes to thank Florian Luca, Harald Niederrei... |

6 |
On the average number of divisors of the Euler function, Publ
- Luca, Pomerance
(Show Context)
Citation Context ...y the question about the behaviour of τ(λ(ℓ)) becomes of interest (where τ(k) is the number of all integer positive divisors of k ≥ 1). Partially motivated by this relations, F. Luca and C. Pomerance =-=[55]-=- obtained tight bounds on the average value of τ(λ(ℓ)). 3.4 Average additive order V. I. Arnold [5] also asks about the average period Q(ℓ) of the map x ↦→ x+a in Zℓ taken over all a ∈ Zℓ. This functi... |

6 |
Dynamical systems generated by rational functions
- Niederreiter, Shparlinski
(Show Context)
Citation Context ...l systems generated by general non-linear transformations x ↦→ f(x) where f is a rational function, over a finite field or a residue ring, have been extensively studied in the literature as well, see =-=[20, 21, 61, 62, 63]-=- and the references therein. As in the case of repeated powering all these results indicate that if the orbit is long enough then its elements are uniformly distributed. On the other hand, these resul... |

5 |
On the Distribution of the Power Generator modulo a Prime Power
- Friedlander, Hansen, et al.
- 2004
(Show Context)
Citation Context |

5 |
A note on polynomial congruences
- Huxley
- 1981
(Show Context)
Citation Context ...p n − 1) | m = 1, . . ., M} and by ψ(y) the multiplicity of y ∈ Y (that is, the number of m = 1, . . ., M with y ≡ f(m) (mod p n − 1)). In particular, #Y ≤ M and by the famous Nagell–Ore theorem (see =-=[43]-=- for its strongest known form) we have Ψ = max y∈Y |ψ(y)| = po(1) . We derive from Lemma 2.1 that max γ∈IF ∗ pn ∑ M∑ ( ) f(m) ep Tr (γϑ ) ∣ ∣ ∣∣∣ ≤ p 9n/8+o(1) M 3/4 , ϑ∈Θ m=1 10 ) .which implies the... |

4 | Double exponential sums related to Diffie-Hellman distributions
- Garaev
(Show Context)
Citation Context ...alogues of our results hold for an arbitrary ϑ, not necessarily a primitive root, provided the multiplicative order of ϑ is large enough. For more general formulations of Lemma 2.1 and Lemma 2.2, see =-=[33, 35]-=- and [27], respectively. Analogues of the bound (9) and Lemmas 2.1 and 2.2 are also known for residue rings Zℓ, see [45, 46, 52] and [28, 29], respectively. Thus one can study orbits of ϑ m in residue... |

4 |
On the linear complexity and multidimensional distribution of congruential generators over elliptic curves’, Designs, Codes and Cryptography
- Hess, Shparlinski
- 2005
(Show Context)
Citation Context ... matrix has been considered in [2]. Using the results and methods of [38], one can prove various uniformity of distribution properties of orbits of such dynamical systems. Accordingly, the results of =-=[7, 41, 50]-=- can be used to obtain similar statements for analogues of the above dynamical systems on elliptic curves over finite fields. Acknowledgements The author wishes to thank Florian Luca, Harald Niederrei... |

4 |
Discrepancy estimates for sets with small boundary
- Laczkovich
- 1995
(Show Context)
Citation Context ....21 of [19]. The link between the distribution in aligned boxes and arbitrary regions is given by the results of H. Niederreiter and J. M. Wills [64] and their more recent refinement of M. Laczkovich =-=[49]-=-. Surprisingly enough, the essentially tautological link between the distribution in arbitrary regions and ergodic properties has never been exploited in a systematic way, although it definitely deser... |

4 |
Some mean values related to average multiplicative orders of elements in finite fields
- Luca
(Show Context)
Citation Context ... adjustments, may shed light on many issues risen by V. I. Arnold in [1, 3, 5, 6]. We also remark that a dual question about the the average value ˜T(ℓ) = 1 ϕ(ℓ) ℓ∑ g=1 gcd(g,ℓ)=1 tg(ℓ) is studied in =-=[37, 54, 56]-=-. In particular, in [56] one can also find various upper and lower bounds on ˜ T(ℓ), while its behaviour on special sequences is considered in [37, 54]. It is well known that if an integer g > 1 is fi... |

4 | Average multiplicative orders of elements modulo n
- Luca, Shparlinski
(Show Context)
Citation Context ... adjustments, may shed light on many issues risen by V. I. Arnold in [1, 3, 5, 6]. We also remark that a dual question about the the average value ˜T(ℓ) = 1 ϕ(ℓ) ℓ∑ g=1 gcd(g,ℓ)=1 tg(ℓ) is studied in =-=[37, 54, 56]-=-. In particular, in [56] one can also find various upper and lower bounds on ˜ T(ℓ), while its behaviour on special sequences is considered in [37, 54]. It is well known that if an integer g > 1 is fi... |

4 |
Design and analysis of nonlinear pseudorandom number generators
- Niederreiter
(Show Context)
Citation Context ...l systems generated by general non-linear transformations x ↦→ f(x) where f is a rational function, over a finite field or a residue ring, have been extensively studied in the literature as well, see =-=[20, 21, 61, 62, 63]-=- and the references therein. As in the case of repeated powering all these results indicate that if the orbit is long enough then its elements are uniformly distributed. On the other hand, these resul... |

4 |
Exponential sums for nonlinear recurring sequences
- Niederreiter, Winterhof
(Show Context)
Citation Context ...l systems generated by general non-linear transformations x ↦→ f(x) where f is a rational function, over a finite field or a residue ring, have been extensively studied in the literature as well, see =-=[20, 21, 61, 62, 63]-=- and the references therein. As in the case of repeated powering all these results indicate that if the orbit is long enough then its elements are uniformly distributed. On the other hand, these resul... |

3 |
Number-theoretical turbulence in Fermat–Euler arithmetics and large Young diagrams geometry statistics
- Arnold
(Show Context)
Citation Context ...45, 46, 52, 59, 60]. Here we provide a brief guide to the literature and demonstrate that many existing techniques are suitable for studying these questions and in fact imply that some conjectures of =-=[1, 3, 5, 6]-=-, based on numerical calculations, need some further adjustments. Throughout this section, the implied constants in the Landau symbol ‘O’ and in the Vinogradov symbols ‘≪’ and ‘≫’ may occasionally, wh... |

3 |
Double character sums over elliptic curves and finite fields
- Banks, Friedlander, et al.
(Show Context)
Citation Context |