On a conjecture of George Beck
نویسندگان
چکیده
منابع مشابه
On the Beck-Fiala Conjecture for Random Set Systems
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability O( √ t log t); and ...
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Beck and Fiala conjectured in 1981 that for any set system S of maximum degree t on a nite ground set X, a coloring : X ! f?1; +1g exists such that j(S)j = O(p t) holds for all S 2 S, where (S) = P x2S (X). We prove a weaker statement, namely that for any xed p 1, a coloring exists such that the pth degree average of j(S)j over S 2 S is O(p t). The result also holds if each set is assigned a no...
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Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2018
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042118500392