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5 Bulletin of the Polish Academy of Sciences. Mathematics

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Exponential sums with Farey fractions

100%

Shparlinski I. E.

2009

tom Vol. 57, no 2

101-107

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set F(N) = {s/r : r, s ∈ Z, gcd(r, s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r, m) = 1 are considered).

On the Euler function on differences between the coordinates of points on modular hyperbolas

100%

Shparlinski I. E.

2008

tom Vol. 56, no 1

1-7

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola Ηa,p(X,Y) = {(x,y) : x,y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. We give asymptotic formulas for the average values (x,y)∈ ... [wzór] with the Euler function φ(k) on the difference between the components of points of Ηa,p(X,Y).

Visible points on curves over finite fields

63%

Shparlinski I. E. , Voloch J. F.

2007

tom Vol. 55, no 3

193-199

For a prime p and an absolutely irreducible modulo p polynomial ƒ(U, V) L 1\U, V] we obtain an asymptotic formula for the number of solutions to the congruence ƒ(x,y) = a (modp) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

On Fully Split Lacunary Polynomials in Finite Fields

63%

Bibak K. , Shparlinski I. E.

2011

tom Vol. 59, no 3

197--202

We estimate the number of possible degree patterns of k-lacunary polynomials of degree t< p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.

Visible points on modular exponential curves

63%

Chan T. H. , Shparlinski I. E.

2010

tom Vol. 58, no 1

17-22

We obtain an asymptotic formula for the number of visible points (x,y). that is, with gcd(x, y) = 1, which lie in the box [1, U] x [1,V] and also belong to the exponential modular curves y ≡ agx (mod p). Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.