Narrow Results

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 and p^ε < |B|, |C| < |A| < p^1-ε, then |A + B| + |A · C| > p^{δ (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 to satisfies |F(A,B)| > p^β for some β = β (α) > α whenever and |A| ~ |B|~ p^α, 0 < α < 1. The exponential sum ∑_{x∈ A,y∈B}ε_p(axy+bx²y²), ab ≠ 0 (mod p), may be estimated nontrivially for arbitrary sets satisfying |A|, |B| > p^ρ 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 p^ε < |A| < p^1-ε, then always |A + A|+|A^-1 + A^-1| > p^δ(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]).

We introduce a class of functions that generalize the epoch-making series of Poincaré and Petersson. Our "uninhibited Poincaré series" permits both a complex weight and an arbitrary multiplier system that is independent of the weight. In this initial paper we provide their Fourier expansions, as well as their modular behavior. We show that they are modular integrals that possess interesting periods. Moreover, we establish with relative ease that they "almost never" vanish identically. Along the way we present a seemingly unknown historical truth concerning Kloosterman sums, and also an alternative approach to Petersson's factor systems. The latter depends upon a simple multiplication rule.

We obtain new upper bounds for the mean squared modulus of sums ∑_n∈ℕA_nχ(n), where the sequence (A_n) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (c_m) satisfying c_m = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has A_n = ∑_{uvm = n} α(u)β(v)c_m (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log_q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

In this paper, we study Kloosterman sums twisted by multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζ_m = exp(2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum

We give an alternative proof of a recent result by T. D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on ${({𝔽}_p)}^2$. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

In this paper, we consider the congruence equation $\overline{a}+\overline{b}\equiv \overline{c}(modp)$ with $1\le a,b,c\le H$ where $\overline{x}$ denotes the multiplicative inverse of $x(modp)$. We prove that its number of solutions is asymptotic to $H^3/p$ when $H>p^{2/3+o(1)}$ by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when $H<p^{1/2}\sqrt{logp}$, the number of solutions has order of magnitude at least $HlogH$. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet $L$-functions at $s=1$.

Let $q,m,n$ be any integers with $q\ge 3$, and $\chi$ be a Dirichlet character modulo $q$. The general Kloosterman sums $K(m,n,\chi ;q)$ are defined as follows:

In this paper, we consider the congruence equation $\overline{a}+\overline{b}\equiv \overline{c}$ (mod p) with 1 ≤ a, b, c ≤ H where $\overline{x}$ denotes the multiplicative inverse of x (mod p). We prove that its number of solutions is asymptotic to H³/p when H > p^2/3+o(1) by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when $H<p^{1/2}\sqrt{\log p}$, the number of solutions has order of magnitude at least H log H. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet L-functions at s = 1.