scholarly journals A sum–product theorem in matrix rings over finite fields

2019 ◽  
Vol 357 (10) ◽  
pp. 766-770 ◽  
Author(s):  
Thang Pham
Keyword(s):  
Finite Fields ◽  
Product Theorem ◽  
Matrix Rings

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-32 ◽  
Author(s):  
J. BOURGAIN
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Functions ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
Product Theorem ◽  
Short Intervals ◽  
Prime Fields ◽  
Product Sets ◽  
Mod P

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε < |B|, |C| < |A| < p1-ε, then |A + B| + |A · C| > pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| > pβ for some β = β (α) > α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 < α < 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| > pρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε < |A| < p1-ε, then always |A + A|+|A-1 + A-1| > pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

Unit-graphs and special unit-digraphs on matrix rings

2018 ◽  
Vol 30 (6) ◽  
pp. 1397-1412 ◽  
Author(s):  
Yeşi̇m Demiroğlu Karabulut
Keyword(s):  
Finite Fields ◽  
Spectral Properties ◽  
Classical Result ◽  
Character Sums ◽  
Kloosterman Sums ◽  
Finite Ring ◽  
Type Result ◽  
Matrix Rings ◽  
Special Unit ◽  
Odd Order

Abstract We use the unit-graphs and the special unit-digraphs on matrix rings to show that every {n\times n} nonzero matrix over {{\mathbb{F}}_{q}} can be written as a sum of two {\operatorname{SL}_{n}} -matrices when {n>1} . We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and we prove that if X is a subset of {\operatorname{Mat}_{2}({\mathbb{F}}_{q})} with size {\lvert X\rvert>2q^{3}\sqrt{q}/(q-1)} , then X contains at least two distinct matrices whose difference has determinant α for any {\alpha\in{\mathbb{F}}_{q}^{\ast}} . Using this result, we also prove a sum-product type result: if {A,B,C,D\subseteq{\mathbb{F}}_{q}} satisfy {\sqrt[4]{\lvert A\rvert\lvert B\rvert\lvert C\rvert\lvert D\rvert}=\Omega(q^{% 0.75})} as {q\rightarrow\infty} , then {(A-B)(C-D)} equals all of {{\mathbb{F}}_{q}^{\ast}} . In particular, if A is a subset of {{\mathbb{F}}_{q}} with cardinality {\lvert A\rvert>\frac{3}{2}q^{3/4}} , then the subset {(A-A)(A-A)} equals all of {{\mathbb{F}}_{q}} . We derive some identities involving character sums of the entries of {2\times 2} matrices over finite fields. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields
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