Towards a problem of Cohn

Journal of Number Theory, 2000

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.

Journal of Algebra, 2021

Chebotarev proved that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We establish these results via a generalization of the Biro-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. We show that our result is best possible and in some cases is stronger than that of Biro-Meshulam-Tao. Some of these results hold in certain circumstances for non-prime fields; Gauss sums play a central role in such investigations.

Pacific Journal of Mathematics, 1970

Let p denote an odd prime. The following three identities (transformation formulae) involving the Legendre symbol ί-j are known to be valid for any complex-valued function F defined on the integers, which is periodic with period p: Σ F(x) + Σ (~V(χ) = Σ *W > x=0 x = 0 \ P / x = 0 v-i p-i //y.2 An\ P-i / n \ Σ F(x) + Σ 1^-^)F(X) = Σ ^(* +-I a 2-Λ We consider a general class of transformation formulae, which includes the above examples. Let p denote a fixed odd prime and let GF(p) denote the Galois field with p elements. If X denotes an indeterminate we let Θ[X] = \θ{X) = aX 2 AX' + BX+C ' ' ' ' ' (aC-cAf-(aB-bA)φC-cB) Φ θl and Φ[X] = {φ(X) = qX 2 + rX+ s\g,r,se GF(p), r*-Aqs Φ 0}. Corresponding to any element θ(X)e6[X] (often just written θeθ) we define Θ*(X) = DX 2 + AX + d , where D = B 2 ~ 4AC, J-4αC-265 + 4cA, d-δ 2-4αc. It is clear that Θ*(X) e Φ[X] as Δ 2-4Dd = 16{(αC-cA) 2-(α£-6A)(6C-cB)} Φ 0. For any element ψ(X) e Φ[X] (often just written φeΦ) its value at #eGF(;p) is just φ(x) = qx 2 + ra; 4se GF(p). For any element ΰ(X) eθ[X], θ(x) will be defined provided Ax 2 + Bx + C Φ 0 and its value is *(*) = T\ + ^ + G r, = (aχ2 + b x + c)(Aχ2 + B x + C)" 1 Aα; 2 + 5ίi? + C 559 560 KENNETH S. WILLIAMS Throughout this paper whenever we write Σ the summation is takem X over all x e GF(p). If we write Σ' the summation is over all x e GF(p) X for which the summand is defined. Further we let c^ denote the complex number field and we denote by ^" the set of all functions with domain GF(p) and range g^. The particular function χ e j^~ defined for any x e GF(p)

Publicationes Mathematicae Debrecen, 1995

2010

We consider the families of finite Abelian groups Z/pZ × Z/pZ, Z/p 2 Z and Z/pZ × Z/qZ for p, q two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality k while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of k in the third case. Such equality cases were previously known when k divides the cardinality of the group, or for groups Z/pZ.

Mathematical and Computer Modelling, 1996

Proceedings of the London Mathematical Society, 1973

ArXiv, 2018

Chebotarev's theorem says that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We then establish a generalization of the Biro-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. We show that our result is best possible and in some cases is stronger than that of Biro-Meshulam-Tao. Some of these results hold in certain circumstances for non-prime fields; Gauss sums play a central role in such investigations.

Acta Mathematica Hungarica, 2020

Let K be an imaginary quadratic number field and OK be its ring of integers. We show that, if the arithmetic functions f, g : OK → C both have level of distribution ϑ for some 0 < ϑ ≤ 1/2 then the Dirichlet convolution f * g also has level of distribution ϑ. As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.

arXiv (Cornell University), 2022

Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [8] about the size of this upper bound.