The Erdös-Falconer Distance Problem on the Unit Sphere in Vector Spaces Over Finite Fields
نویسنده
چکیده
Hart, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erdös-Falconer distance conjecture holds for subsets of the unit sphere in Fdq . In this note, we give a graph theoretic proof of this result.
منابع مشابه
Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering Fq, the finite field with q elements,...
متن کاملThe Erdös--Falconer Distance Problem, Exponential Sums, and Fourier Analytic Approach to Incidence Theorems in Vector Spaces over Finite Fields
We study the Erdös/Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson/Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a non-trivial rang...
متن کاملErdos Distance Problem in Vector Spaces over Finite Fields
The classical Erdos distance conjecture says that the number of distance determined by N points in Rd, d ≥ 2, is at least CN 2 d − . We shall discuss this problem in vector spaces over finite fields where some interesting number theory comes into play. A connection with the continuous analog of the Erdos distance conjecture, the Falconer distance conjecture will be also be established. Universi...
متن کاملAdditive Energy and the Falconer Distance Problem in Finite Fields
We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of the Fourier transform on given sets. As an application of our observation, we find sufficient conditions on sets where the Falconer distance conjecture for fin...
متن کاملThe generalized Erdös-Falconer distance problems in vector spaces over finite fields
In this paper we study the generalized Erdös-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical dis...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Discrete Math.
دوره 25 شماره
صفحات -
تاریخ انتشار 2011