Pair correlation for quadratic polynomials mod 1
نویسندگان
چکیده
منابع مشابه
The Correlation Between Parity and Quadratic Polynomials Mod 3
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3: One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2OðnÞ: This is the first result of this type for general mod 3 subcircuits with ANDs of fan-i...
متن کاملPair Correlation Densities of Inhomogeneous Quadratic Forms
Under explicit diophantine conditions on (α, β) ∈ R2, we prove that the local two-point correlations of the sequence given by the values (m − α)2 + (n−β)2, with (m,n) ∈ Z2, are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppen...
متن کاملThe Correlation Between Parity and Quadratic Polynomials
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2. This is the first result of this type for general mod 3 subcircuits with ANDs of fan-in gr...
متن کاملPair Correlation Densities of Inhomogeneous Quadratic Forms Ii
Abstract. Denote by ‖ · ‖ the euclidean norm in R. We prove that the local pair correlation density of the sequence ‖m− α‖k, m ∈ Z, is that of a Poisson process, under diophantine conditions on the fixed vector α ∈ R: in dimension two, vectors α of any diophantine type are admissible; in higher dimensions (k > 2), Poisson statistics are only observed for diophantine vectors of type κ < (k − 1)/...
متن کاملCorrection to “pair Correlation Densities of Inhomogeneous Quadratic Forms, Ii”
Proof One follows the argument in [1, pp. 432 – 433] (cf. also [2, Sec. 9]) to show that for each fixed badly approximable (k − 2)-tuple (α1, . . . , αk−2) there is a set of second Baire category of (αk−1, αk) ∈ T2 such that conditions (i), (ii), and (iii) of Theorem 1.7 hold for α = (α1, . . . , αk). Because the set of badly approximable (k − 2)-tuples is dense in Tk−2, and the set of second B...
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ژورنال
عنوان ژورنال: Compositio Mathematica
سال: 2018
ISSN: 0010-437X,1570-5846
DOI: 10.1112/s0010437x17008028