Let m be a positive integer and L the lattice of points in Z 2 defined by the congruence ax+by ≡ ... more Let m be a positive integer and L the lattice of points in Z 2 defined by the congruence ax+by ≡ 0 (mod m) where (a, b, m) = 1. We prove that if L contains a point in the region R m = {(x, y) : 0 ≤ |x| ≤ √ m, 0 ≤ |y| ≤ √ m, |x| + 2|y| ≥ √ m, 2|x| + |y| ≥ √ m} with relatively prime coordinates then the points of L are well distributed in the sense that for any integers A, B, L contains a point (x, y) with |A − x| ≤ √ m and |B − y| ≤ √ m. We then apply this result to show that for any integers A, B, C, D and c the quadratic congruence xy+uv ≡ c (mod m) has a solution with |x−A| ≤ √ m, |y −B| ≤ (1+[δ/2]) √ m, |u−C| ≤ √ m and |v − D| ≤ √ m + [δ/2], where δ is the maximum gap between reduced residues (mod m).
Let Q(x) be a quadratic form over Z in n variables, p be an odd prime and âx â = maxi |xi|. A... more Let Q(x) be a quadratic form over Z in n variables, p be an odd prime and âx â = maxi |xi|. A solution of the congruence Q(x) â¡ 0 (mod p2) is said to be nontrivial if p xi for some i. We prove that if this congruence has a nontrivial solution, then it has a nontrivial solution with âx â ⤠p. We also give estimates on the number of small nontrivial solutions of the congruence and show that there exists a set of n linearly independent nontrivial solutions of size âx â ⤠(2n+1+1)p, provided that n ⥠4 is even and Q(x) is nonsingular (mod p).
Functiones et Approximatio Commentarii Mathematici, 2017
We prove that for k ≥ 2, 0 < ε < 1 k(k−1) , n > k−1 ε , prime p > P (ε, k), and integers c, a i ,... more We prove that for k ≥ 2, 0 < ε < 1 k(k−1) , n > k−1 ε , prime p > P (ε, k), and integers c, a i , with p a i , 1 ≤ i ≤ n, there exists a solution x to the congruence n i=1 a i x k i ≡ c mod p in any cube B of side length b ≥ p 1 k +ε. Various refinements are given for smaller n and for cubes centered at the origin.
For n ≥ 3 we obtain an improved estimate for the generalized Heilbronn sum p−1 x=1 e p n (yx p n−... more For n ≥ 3 we obtain an improved estimate for the generalized Heilbronn sum p−1 x=1 e p n (yx p n−1), and use it to show that any interval I of points in Z p n of length |I| p 1.825 for n = 2, |I| p 2.959 for n = 3, |I| ≥ p n−3.269(34/151) n +o(1) for n ≥ 4, contains a (p − 1)-th root of unity. As a consequence, we derive an improved estimate for the Lind-Lehmer constant for the abelian group Z n p , and improved estimates for Fermat quotients.
Proceedings of the American Mathematical Society, 1988
We construct n n -dimensional versions of the Beurling and Selberg majorizing and minorizing func... more We construct n n -dimensional versions of the Beurling and Selberg majorizing and minorizing functions and use them to prove results on trigonometric approximation and to prove an n n -dimensional version of the Erdös-Turán inequality. Finally, an application is given to counting solutions of polynomial congruences.
Proceedings of the American Mathematical Society, 1989
Let C ( x ) C({\mathbf {x}}) be a cubic form in n n variables over Z {\mathbf {Z}} and p p be a p... more Let C ( x ) C({\mathbf {x}}) be a cubic form in n n variables over Z {\mathbf {Z}} and p p be a prime. Then for 0 > σ > 2 3 0 > \sigma > \frac {2}{3} the congruence C ( x ) ≡ 0 ( mod p ) C({\mathbf {x}}) \equiv 0(\bmod p) has a nonzero solution x x with max | x i | ≪ p 1 / 3 + σ \max \left | {{x_i}} \right | \ll {p^{1/3 + \sigma }} , provided that n > 8 / σ n > 8/\sigma , (where the constant in the ≪ \ll depends on n n and σ \sigma ).
Cohn's problem on character sums (see [6], p. 202) asks whether a multiplicative character on a f... more Cohn's problem on character sums (see [6], p. 202) asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let f be a map from a finite field F to the complex plane such that f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0. In this paper we show that if for all a, b ∈ F * , we have (q − 1) α∈F f (bα)f (α + a) = − α∈F f (bα)f (α), then f is a multiplicative character of F. We also prove that if F is a prime field and f is a real valued function on F with f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0, then α∈F f (α)f (α + a) = −1 for all a = 0 if and only if f is the Legendre symbol. These results partially answer Cohn's problem.
For any integers al,a2,a3,a4 and c with ala2a3a4 ~ 0(rood p), this paper shows that there exists ... more For any integers al,a2,a3,a4 and c with ala2a3a4 ~ 0(rood p), this paper shows that there exists a solution X = (xz,z2,x3,z4) 9 Z 4 of the congruence alz~ + a2x 2 + a3z~ + a4x24 = c(mod p) such that IlXll = max{Ixl[, i=21, [x3[, 1~4[} << v~logp.
Transactions of the American Mathematical Society, 1986
Let F q {\mathbb {F}_q} be the finite field in q = p f q = {p^f} elements, F _ ( x _ ) \underline... more Let F q {\mathbb {F}_q} be the finite field in q = p f q = {p^f} elements, F _ ( x _ ) \underline F (\underline x ) be a k k -tuple of polynomials in F q [ x 1 , … , x n ] {\mathbb {F}_q}[{x_1}, \ldots ,{x_n}] , V V be the set of points in F q n \mathbb {F}_q^n satisfying F _ ( x _ ) = 0 _ \underline F (\underline x ) = \underline 0 and S S , T T be any subsets of F q n \mathbb {F}_q^n . Set ϕ ( V , 0 _ ) = | V | − q n − k \phi (V,\underline 0 ) = |V| - {q^{n - k}} , \[ ϕ ( V , y _ ) = ∑ x _ ∈ V e ( 2 π i p Tr ( x _ ⋅ y _ ) ) for y _ ≠ 0 _ , \phi (V,\underline y ) = \sum \limits _{\underline x \in V} {e\left ( {\frac {{2\pi i}} {p}\operatorname {Tr} (\underline x \cdot \underline y )} \right )\quad {\text {for}}\;\underline y \ne \underline 0 ,} \] and Φ ( V ) = max y _ | ϕ ( V , y _ ) | \Phi (V) = {\max _{\underline y }}|\phi (V,\underline y )| . We use finite Fourier series to show that ( S + T ) ∩ V (S + T) \cap V is nonempty if | S | | T | > Φ 2 ( V ) q 2 k |S||T| > {\Ph...
For a polynomial F = F (x) ∈ C[x], the logarithmic Mahler measure of F is defined by m(F) = ∫ 1 0... more For a polynomial F = F (x) ∈ C[x], the logarithmic Mahler measure of F is defined by m(F) = ∫ 1 0 log |F (e2piix) | dx. Lind viewed [0, 1] as the group R/Z and extended the concept of the Mahler measure to arbitrary compact abelian groups. In this thesis we study the additive group G:= Zp×Zp× · · ·×Zp, where Zp = Z/(p). For F ∈ Z[x1, x2,..., xn] the logarithmic Lind-Mahler measure of F over G is given by mG(F):= 1 pn p−1∑ j1=0 p−1∑ j2=0 p−1∑ jn=0 log |F (e2piij1/p, e2piij2/p,..., e2piijn/p)|. Analogous to the classic Lehmer problem one can ask what is the smallest nonzero Lind-Mahler measure, that is, what is the value of the Lind-Lehmer constant of G defined as follows: λ(G): = inf{mG(F) | F ∈ Z[x1, x2,..., xn],mG(F)> 0}. Lind found the Lind-Lehmer constant for many groups of the form Zn. Lind also conjec-tured the value of λ(Zn2). Here we verify the Lind conjecture and more generally evaluate λ(Znp) for arbitrary primes p and n ∈ N, obtaining for any p ≥ 3, λ(Znp) = 1 pn log(Mn...
International Journal of Applied and Computational Mathematics, 2014
A doubling transformation ψ(x) = x − λ − b x−μ , b, λ, μ ∈ R, b > 0, has the property that for an... more A doubling transformation ψ(x) = x − λ − b x−μ , b, λ, μ ∈ R, b > 0, has the property that for any absolutely integral function F(x) on R we have ∞ −∞ F(ψ(x)) dx = ∞ −∞ F(x) dx. Compositions of doubling transformations also satisfy this integral invariance property. In this paper we give criteria for determining when a given rational function is a composition of two or more doubling transformations, and use this criteria for giving explicit families of such transformations such as (x−a)(x+a 2)(x−a 3)(x+a 4) x(x−a 2)(x+a 3) , for a > 1; (x 2 −a 2)(x 2 −a 8) x(x+a 2)(x−a 3) , for a > 1; and (x 2 −a 2)(x 2 −b 2) x(x 2 −ab) , for 0 < a < b. Keywords Doubling transformations • Invariant integrals Mathematics Subject Classification 26A42 • 26A33 • 26A48 • 26C15 with λ, μ, b real numbers with b > 0, called doubling transformations, have the interesting property that for any absolutely integrable function F(x) on R, we have ∞ −∞
where p is a prime power, epm(·) is the additive character epm(x) = e m and χ is a multiplicative... more where p is a prime power, epm(·) is the additive character epm(x) = e m and χ is a multiplicative character (mod p). The goals of this paper are threefold; first, to point out the similarity between exponential sums over finite fields and exponential sums over residue class rings (mod p) with m ≥ 2; second, to show how mixed exponential sums can be reduced to pure exponential sums when m ≥ 2 and third, to make a thorough review of the formulae and upper bounds that are available for such sums. Included are some new observations and consequences of the methods we have developed as well as a number of open questions, some very deep and some readily accessible, inviting the reader to a further investigation of these sums.
We prove an analogue of the Cauchy-Davenport Theorem and Chowla's Theorem for sum sets in a gener... more We prove an analogue of the Cauchy-Davenport Theorem and Chowla's Theorem for sum sets in a general abelian group and give an application to diagonal congruences, establishing a best possible estimate for the distribution of solutions of a diagonal congruence n i=1 a i x k i ≡ c (mod q) with an arbitrary modulus.
For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f (... more For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f (x) = Ax k mod p cannot map a residue class mod n to just one residue class mod n once p is sufficiently large, other than the maps f (x) = ±x mod p when n is even and f (x) = ±x or ±x (p+1)/2 mod p when n is odd.
Mathematical Proceedings of the Cambridge Philosophical Society, 2010
We obtain a number of new bounds for exponential sums of the type S(χ, f) = P p−1 x=1 χ(x)ep(f (x... more We obtain a number of new bounds for exponential sums of the type S(χ, f) = P p−1 x=1 χ(x)ep(f (x)), with p a prime, f (x) = P r i=1 a i x k i , a i , k i ∈ Z, 1 ≤ i ≤ r and χ a multiplicative character (mod p). The bounds refine earlier Mordell-type estimates and are particularly effective for polynomials in which a certain number of the k i have a large gcd with p − 1. For instance, if f (x) = P m i=1 a i x k i + g(x d) with d|(p − 1) then |S(χ, f)| ≤ p (k 1 • • • km) 1 m 2 /d 1 2m. If f (x) = ax k + h(x d) with d|(p − 1) and (k, p − 1) = 1 then |S(χ, f)| ≤ p/ √ d, and if f (x) = ax k + bx −k + h(x d) with d|(p − 1) and (k, p − 1) = 1 then |S(χ, f)| ≤ p/ √ d + √ 2p 3/4 .
We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding ... more We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length |I| q 1 2 +. Similar results are obtained for the distribution of ordered pairs (x, x −1) with x a primitive λroot, and for the number of primitive λ-roots satisfying inequalities such as |x − x −1 | ≤ B.
Let m be a positive integer and L the lattice of points in Z 2 defined by the congruence ax+by ≡ ... more Let m be a positive integer and L the lattice of points in Z 2 defined by the congruence ax+by ≡ 0 (mod m) where (a, b, m) = 1. We prove that if L contains a point in the region R m = {(x, y) : 0 ≤ |x| ≤ √ m, 0 ≤ |y| ≤ √ m, |x| + 2|y| ≥ √ m, 2|x| + |y| ≥ √ m} with relatively prime coordinates then the points of L are well distributed in the sense that for any integers A, B, L contains a point (x, y) with |A − x| ≤ √ m and |B − y| ≤ √ m. We then apply this result to show that for any integers A, B, C, D and c the quadratic congruence xy+uv ≡ c (mod m) has a solution with |x−A| ≤ √ m, |y −B| ≤ (1+[δ/2]) √ m, |u−C| ≤ √ m and |v − D| ≤ √ m + [δ/2], where δ is the maximum gap between reduced residues (mod m).
Let Q(x) be a quadratic form over Z in n variables, p be an odd prime and âx â = maxi |xi|. A... more Let Q(x) be a quadratic form over Z in n variables, p be an odd prime and âx â = maxi |xi|. A solution of the congruence Q(x) â¡ 0 (mod p2) is said to be nontrivial if p xi for some i. We prove that if this congruence has a nontrivial solution, then it has a nontrivial solution with âx â ⤠p. We also give estimates on the number of small nontrivial solutions of the congruence and show that there exists a set of n linearly independent nontrivial solutions of size âx â ⤠(2n+1+1)p, provided that n ⥠4 is even and Q(x) is nonsingular (mod p).
Functiones et Approximatio Commentarii Mathematici, 2017
We prove that for k ≥ 2, 0 < ε < 1 k(k−1) , n > k−1 ε , prime p > P (ε, k), and integers c, a i ,... more We prove that for k ≥ 2, 0 < ε < 1 k(k−1) , n > k−1 ε , prime p > P (ε, k), and integers c, a i , with p a i , 1 ≤ i ≤ n, there exists a solution x to the congruence n i=1 a i x k i ≡ c mod p in any cube B of side length b ≥ p 1 k +ε. Various refinements are given for smaller n and for cubes centered at the origin.
For n ≥ 3 we obtain an improved estimate for the generalized Heilbronn sum p−1 x=1 e p n (yx p n−... more For n ≥ 3 we obtain an improved estimate for the generalized Heilbronn sum p−1 x=1 e p n (yx p n−1), and use it to show that any interval I of points in Z p n of length |I| p 1.825 for n = 2, |I| p 2.959 for n = 3, |I| ≥ p n−3.269(34/151) n +o(1) for n ≥ 4, contains a (p − 1)-th root of unity. As a consequence, we derive an improved estimate for the Lind-Lehmer constant for the abelian group Z n p , and improved estimates for Fermat quotients.
Proceedings of the American Mathematical Society, 1988
We construct n n -dimensional versions of the Beurling and Selberg majorizing and minorizing func... more We construct n n -dimensional versions of the Beurling and Selberg majorizing and minorizing functions and use them to prove results on trigonometric approximation and to prove an n n -dimensional version of the Erdös-Turán inequality. Finally, an application is given to counting solutions of polynomial congruences.
Proceedings of the American Mathematical Society, 1989
Let C ( x ) C({\mathbf {x}}) be a cubic form in n n variables over Z {\mathbf {Z}} and p p be a p... more Let C ( x ) C({\mathbf {x}}) be a cubic form in n n variables over Z {\mathbf {Z}} and p p be a prime. Then for 0 > σ > 2 3 0 > \sigma > \frac {2}{3} the congruence C ( x ) ≡ 0 ( mod p ) C({\mathbf {x}}) \equiv 0(\bmod p) has a nonzero solution x x with max | x i | ≪ p 1 / 3 + σ \max \left | {{x_i}} \right | \ll {p^{1/3 + \sigma }} , provided that n > 8 / σ n > 8/\sigma , (where the constant in the ≪ \ll depends on n n and σ \sigma ).
Cohn's problem on character sums (see [6], p. 202) asks whether a multiplicative character on a f... more Cohn's problem on character sums (see [6], p. 202) asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let f be a map from a finite field F to the complex plane such that f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0. In this paper we show that if for all a, b ∈ F * , we have (q − 1) α∈F f (bα)f (α + a) = − α∈F f (bα)f (α), then f is a multiplicative character of F. We also prove that if F is a prime field and f is a real valued function on F with f (0) = 0, f (1) = 1, and | f (α) |= 1 for all α = 0, then α∈F f (α)f (α + a) = −1 for all a = 0 if and only if f is the Legendre symbol. These results partially answer Cohn's problem.
For any integers al,a2,a3,a4 and c with ala2a3a4 ~ 0(rood p), this paper shows that there exists ... more For any integers al,a2,a3,a4 and c with ala2a3a4 ~ 0(rood p), this paper shows that there exists a solution X = (xz,z2,x3,z4) 9 Z 4 of the congruence alz~ + a2x 2 + a3z~ + a4x24 = c(mod p) such that IlXll = max{Ixl[, i=21, [x3[, 1~4[} << v~logp.
Transactions of the American Mathematical Society, 1986
Let F q {\mathbb {F}_q} be the finite field in q = p f q = {p^f} elements, F _ ( x _ ) \underline... more Let F q {\mathbb {F}_q} be the finite field in q = p f q = {p^f} elements, F _ ( x _ ) \underline F (\underline x ) be a k k -tuple of polynomials in F q [ x 1 , … , x n ] {\mathbb {F}_q}[{x_1}, \ldots ,{x_n}] , V V be the set of points in F q n \mathbb {F}_q^n satisfying F _ ( x _ ) = 0 _ \underline F (\underline x ) = \underline 0 and S S , T T be any subsets of F q n \mathbb {F}_q^n . Set ϕ ( V , 0 _ ) = | V | − q n − k \phi (V,\underline 0 ) = |V| - {q^{n - k}} , \[ ϕ ( V , y _ ) = ∑ x _ ∈ V e ( 2 π i p Tr ( x _ ⋅ y _ ) ) for y _ ≠ 0 _ , \phi (V,\underline y ) = \sum \limits _{\underline x \in V} {e\left ( {\frac {{2\pi i}} {p}\operatorname {Tr} (\underline x \cdot \underline y )} \right )\quad {\text {for}}\;\underline y \ne \underline 0 ,} \] and Φ ( V ) = max y _ | ϕ ( V , y _ ) | \Phi (V) = {\max _{\underline y }}|\phi (V,\underline y )| . We use finite Fourier series to show that ( S + T ) ∩ V (S + T) \cap V is nonempty if | S | | T | > Φ 2 ( V ) q 2 k |S||T| > {\Ph...
For a polynomial F = F (x) ∈ C[x], the logarithmic Mahler measure of F is defined by m(F) = ∫ 1 0... more For a polynomial F = F (x) ∈ C[x], the logarithmic Mahler measure of F is defined by m(F) = ∫ 1 0 log |F (e2piix) | dx. Lind viewed [0, 1] as the group R/Z and extended the concept of the Mahler measure to arbitrary compact abelian groups. In this thesis we study the additive group G:= Zp×Zp× · · ·×Zp, where Zp = Z/(p). For F ∈ Z[x1, x2,..., xn] the logarithmic Lind-Mahler measure of F over G is given by mG(F):= 1 pn p−1∑ j1=0 p−1∑ j2=0 p−1∑ jn=0 log |F (e2piij1/p, e2piij2/p,..., e2piijn/p)|. Analogous to the classic Lehmer problem one can ask what is the smallest nonzero Lind-Mahler measure, that is, what is the value of the Lind-Lehmer constant of G defined as follows: λ(G): = inf{mG(F) | F ∈ Z[x1, x2,..., xn],mG(F)> 0}. Lind found the Lind-Lehmer constant for many groups of the form Zn. Lind also conjec-tured the value of λ(Zn2). Here we verify the Lind conjecture and more generally evaluate λ(Znp) for arbitrary primes p and n ∈ N, obtaining for any p ≥ 3, λ(Znp) = 1 pn log(Mn...
International Journal of Applied and Computational Mathematics, 2014
A doubling transformation ψ(x) = x − λ − b x−μ , b, λ, μ ∈ R, b > 0, has the property that for an... more A doubling transformation ψ(x) = x − λ − b x−μ , b, λ, μ ∈ R, b > 0, has the property that for any absolutely integral function F(x) on R we have ∞ −∞ F(ψ(x)) dx = ∞ −∞ F(x) dx. Compositions of doubling transformations also satisfy this integral invariance property. In this paper we give criteria for determining when a given rational function is a composition of two or more doubling transformations, and use this criteria for giving explicit families of such transformations such as (x−a)(x+a 2)(x−a 3)(x+a 4) x(x−a 2)(x+a 3) , for a > 1; (x 2 −a 2)(x 2 −a 8) x(x+a 2)(x−a 3) , for a > 1; and (x 2 −a 2)(x 2 −b 2) x(x 2 −ab) , for 0 < a < b. Keywords Doubling transformations • Invariant integrals Mathematics Subject Classification 26A42 • 26A33 • 26A48 • 26C15 with λ, μ, b real numbers with b > 0, called doubling transformations, have the interesting property that for any absolutely integrable function F(x) on R, we have ∞ −∞
where p is a prime power, epm(·) is the additive character epm(x) = e m and χ is a multiplicative... more where p is a prime power, epm(·) is the additive character epm(x) = e m and χ is a multiplicative character (mod p). The goals of this paper are threefold; first, to point out the similarity between exponential sums over finite fields and exponential sums over residue class rings (mod p) with m ≥ 2; second, to show how mixed exponential sums can be reduced to pure exponential sums when m ≥ 2 and third, to make a thorough review of the formulae and upper bounds that are available for such sums. Included are some new observations and consequences of the methods we have developed as well as a number of open questions, some very deep and some readily accessible, inviting the reader to a further investigation of these sums.
We prove an analogue of the Cauchy-Davenport Theorem and Chowla's Theorem for sum sets in a gener... more We prove an analogue of the Cauchy-Davenport Theorem and Chowla's Theorem for sum sets in a general abelian group and give an application to diagonal congruences, establishing a best possible estimate for the distribution of solutions of a diagonal congruence n i=1 a i x k i ≡ c (mod q) with an arbitrary modulus.
For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f (... more For a fixed integer n ≥ 2, we show that a permutation of the least residues mod p of the form f (x) = Ax k mod p cannot map a residue class mod n to just one residue class mod n once p is sufficiently large, other than the maps f (x) = ±x mod p when n is even and f (x) = ±x or ±x (p+1)/2 mod p when n is odd.
Mathematical Proceedings of the Cambridge Philosophical Society, 2010
We obtain a number of new bounds for exponential sums of the type S(χ, f) = P p−1 x=1 χ(x)ep(f (x... more We obtain a number of new bounds for exponential sums of the type S(χ, f) = P p−1 x=1 χ(x)ep(f (x)), with p a prime, f (x) = P r i=1 a i x k i , a i , k i ∈ Z, 1 ≤ i ≤ r and χ a multiplicative character (mod p). The bounds refine earlier Mordell-type estimates and are particularly effective for polynomials in which a certain number of the k i have a large gcd with p − 1. For instance, if f (x) = P m i=1 a i x k i + g(x d) with d|(p − 1) then |S(χ, f)| ≤ p (k 1 • • • km) 1 m 2 /d 1 2m. If f (x) = ax k + h(x d) with d|(p − 1) and (k, p − 1) = 1 then |S(χ, f)| ≤ p/ √ d, and if f (x) = ax k + bx −k + h(x d) with d|(p − 1) and (k, p − 1) = 1 then |S(χ, f)| ≤ p/ √ d + √ 2p 3/4 .
We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding ... more We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length |I| q 1 2 +. Similar results are obtained for the distribution of ordered pairs (x, x −1) with x a primitive λroot, and for the number of primitive λ-roots satisfying inequalities such as |x − x −1 | ≤ B.
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