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Analytic Number Theory
 A.M.S Colloquium Publ
"... Key words and phrases. Analytic number theory, distribution of ..."
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Cited by 246 (28 self)
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Key words and phrases. Analytic number theory, distribution of
Erdős distance problem in vector spaces over finite fields
 Transactions of the American Mathematical Society
"... Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F ..."
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Cited by 33 (9 self)
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Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) intermsofthe cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof. 1.
On QuasiOrthogonal Signatures for CDMA Systems
 IEEE Trans. Inform. Theory. [Online]. Available: http://www.ece.utexas.edu/˜rheath/papers/2002/quasicdma
, 2002
"... Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed fo ..."
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Cited by 20 (7 self)
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Codebooks constructed from Welch bound equality sequences are optimal in terms of sum capacity in synchronous CDMA systems. These optimal codebooks depend on both the sequence length as well as the number of active sequences. Thus to maintain optimality a typically different codebook is needed for every possible number of active users. Further, all the sequences need to be reassigned as the number of active users changes. This paper describes and analyzes two promising subclasses of Welch bound equality sequences that have good properties when only a subset of sequences is active. One subclass, based on maximum Welch bound equality sequence sets, has equiangular sequences thus each user experiences the same amount of interference. The interference power depends only on the total number of active users. Another subclass, constructed by concatenating multiple orthonormal basis, comes closer to the Welch bound when not all signatures are active. Optimal unions of orthonormal basis are derived. Performance when subsets of sequences are active is characterized in terms of sumsquared correlation, average interference, and condition number of the Gram matrix.
The analytic rank of J0(q) and zeros of automorphic Lfunctions
"... We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides ..."
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Cited by 16 (11 self)
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We study zeros of the Lfunctions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the Lfunctions at the central critical point s = 1 2 is absolutely bounded. On the Birch and SwinnertonDyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the same order of magnitude as what is expected to be true. 1
Sumproduct Estimates in Finite Fields via Kloosterman Sums
"... We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums. ..."
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Cited by 15 (2 self)
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We establish improved sumproduct bounds in finite fields using incidence theorems based on bounds for classical Kloosterman and related sums.
ErdösFalconer 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 incide ..."
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Cited by 14 (5 self)
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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 nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.
The distribution of the free path lengths in the periodic twodimensional Lorentz gas in the smallscatterer limit
, 2003
"... We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one get ..."
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Cited by 14 (6 self)
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We study the free path length and the geometric free path length in the model of the periodic twodimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the smallscatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term c = 2 − 3ln 2 + 27ζ(3) 2π2 in the asymptotic formula h(T) = −2ln ε + c + o(1) of the KS entropy of the billiard map in this model.
Quantum algorithms for hidden nonlinear structures
 Proc. 48th IEEE Symp. on Found. Comp. Sci., IEEE
"... Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor’s factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden n ..."
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Cited by 14 (4 self)
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Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor’s factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures. 1