On the squares in the set of elements of a finite field with constraints on the coefficients of its basis expansion

R., Gabdullin, Mikhail

doi:10.1134/s000143461701028x

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…An analog for finite fields of the problem on the distribution of automatic sequences and their subsequences was introduced by Dartyge and Sárközy [22]. It has been further investigated in [21,55,57,94,95], see also [20,26,37,77,96].…”

Section: Analogs For Finite Fieldsmentioning

confidence: 99%

On the pseudorandomness of automatic sequences

Mérai

Winterhof

2017

Cryptogr. Commun.

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Section: Analogs For Finite Fieldsmentioning

confidence: 99%

On the pseudorandomness of automatic sequences

Mérai

Winterhof

2017

Cryptogr. Commun.

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“…. , dr ∈ D} was studied in [1,5,8], that is, f (ξ) 'misses' the digits in Fp \ D. It is straightforward to extend these results combining our approach with certain bounds on character sums to estimate the number of ξ ∈ Fq with f (ξ + αi) ∈ {d1β1 + . .…”

Section: Missing Digits and Subsetsmentioning

confidence: 91%

“…Proof of the first part: We have to show that ( 6) is applicable, that is, the polynomial Fa 1 ,...,as (X) defined by (8) is not of the form g(X) p − g(X) + c for any (a1, . .…”

Section: Arbitrary Polynomialsmentioning

confidence: 99%

Normality of the Thue-Morse function for finite fields along polynomial values

Makhul¹,

Winterhof²

2021

Preprint

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Let Fq be the finite field of q elements, where q = p r is a power of the prime p, and (β1, β2, . . . , βr) be an ordered basis of Fq over Fp. Forwe define the Thue-Morse or sum-of-digits function T (ξ) on Fq byxi.For a given pattern length s with 1 ≤ s ≤ q, a subset A = {α1, . . . , αs} ⊂ Fq, a polynomial f (X) ∈ Fq[X] of degree d and a vector c = (c1, . . . , cs) ∈ F s p we putIn this paper we will see that under some natural conditions, the size of T (c, A, f ) is asymptotically the same for all c and A in both cases, p → ∞ and r → ∞, respectively. More precisely, we haveunder certain conditions on d, q and s. For monomials of large degree we improve this bound as well as we find conditions on d, q and s for which this bound is not true. In particular, if 1 ≤ d < p we have the dichotomy that the bound is valid if s ≤ d and fails for some c and A if s ≥ d + 1. The case s = 1 was studied before by Dartyge and Sárközy.

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“…A natural question is to study analog problems in finite fields, see for example [4,5,7,9,12,18,[20][21][22]. Many of these problems can be solved for finite fields although their analogs for integers are actually out of reach.…”

Section: Introductionmentioning

confidence: 99%

On the distribution of the Rudin-Shapiro function for finite fields

Dartyge¹,

Mérai²,

Winterhof³

2020

Preprint

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Let q = p r be the power of a prime p and (β1, . . . , βr) be an ordered basis of Fq over Fp. For ξ = r j=1 xjβj ∈ Fq with digits xj ∈ Fp, we define the Rudin-Shapiro function R on Fq byFor a non-constant polynomial f (X) ∈ Fq[X] and c ∈ Fp we study the number of solutions ξ ∈ Fq of R(f (ξ)) = c. If the degree d of f (X) is fixed, r ≥ 6 and p → ∞, the number of solutions is asymptotically p r−1 for any c. The proof is based on the Hooley-Katz Theorem.

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On the squares in the set of elements of a finite field with constraints on the coefficients of its basis expansion

Cited by 8 publications

References 11 publications

On the pseudorandomness of automatic sequences

On the pseudorandomness of automatic sequences

Normality of the Thue-Morse function for finite fields along polynomial values

On the distribution of the Rudin-Shapiro function for finite fields

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