P. Erdo# s and P. Tura# n, On a problem of Sidon in additive number theory, and some related problems, J. London Math. Soc. 16 (  Home/Search   Document Not in Database   Summary   Related Articles   Check

....g(n; e) Theta(e) whenever e = O(n 3=2 ) In Section 2.3, we give a probabilistic proof for the existence of a set B with property L( Theta(n) O(log 2 n) which gives a near quadratic lower bound for f(n) 2. 2 A Constructive Lower Bound We employ the following result of Erdos and Tur an [11], proved independently by Singer [16] For the sake of completeness, we include the proof given in [11] Lemma 2.5 (Erdos Tur an [11] Given any integer m 0, let T (m) foe 1 ; oe 2 ; oe t g f1; mg be a largest cardinality set such that oe i oe j 6= oe i 0 oe j 0 ....

....of a set B with property L( Theta(n) O(log 2 n) which gives a near quadratic lower bound for f(n) 2. 2 A Constructive Lower Bound We employ the following result of Erdos and Tur an [11] proved independently by Singer [16] For the sake of completeness, we include the proof given in [11]. Lemma 2.5 (Erdos Tur an [11] Given any integer m 0, let T (m) foe 1 ; oe 2 ; oe t g f1; mg be a largest cardinality set such that oe i oe j 6= oe i 0 oe j 0 whenever fi; jg 6= fi 0 ; j 0 g. Then, t = Theta( p m) 7 Proof: It is clear that a larger set ....

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P. Erdos and P. Tur'an, On a problem of Sidon in additive number theory, and on some related problems, Proc. London Math. Soc. 16 (1941), 212--215.

....Examples of asymptotic additive bases are the set N of natural numbers itself and the set f1; 2; 4; 6; 8; g. We are interested in bases for which the representation function is small. Notice that in the previous two examples r(x) can be as large as Cx. We present Erdos s probabilistic proof [8, 9], 14, Ch. 3] that there is an asymptotic basis of order 2 such that c 1 log x r(x) c 2 log x (10) for all sufficiently large x. The ratio of the two absolute constants c 1 and c 2 can be made arbitrarily close to 1. Define the probabilities p x = K Delta log x x 1=2 for the values of ....

....probability space, one faces the obstacle that making the decision whether to put a certain integer m in the set E or not affects the values for r E (x) for all x m, which are of course infinitely many. This problem can be overcome if one looks at the original, slightly different, proof of Erdos [8], which has been stated using counting arguments and not probability. It uses an existential argument on a finite interval at a time and can thus be readily turned into a construction by examining all possible intersections of E with the interval. But the algorithm which we get this way takes time ....

P. Erdos, On a problem of Sidon in additive number theory, Acta Sci. Math. (Szeged), 15 (1953-54), 255-259.

....of representations of x as a b, with a; b 2 E and a b. In what follows C denotes an arbitrary positive constant, not necessarily the same in all its occurences, and N = f1; 2; 3; g denotes the set of all positive integers. The mean value of a random variable X is denoted by EX . Erdos [2, 3] has proved that there is a basis E such that C log x r(x) C log x (1) for all positive integers x (see also [1, p. 106] and [4, Ch. 3] The most widely known proof (in [1, 3, 4] is probabilistic. It is proved that if we let x 2 E with a certain probability p x , independently for all x, then ....

....p x , independently for all x, then the random set E is an asymptotic basis (that is (1) is true eventually) with probability 1. Since the probability space used is infinite, the question of whether such a basis exists which is also computable is not addressed by this proof. The original [2] proof though, which has been stated using counting arguments and not probability, uses an existential argument on a finite interval at a time and can thus be readily turned into a construction by examining all possible intersections of E with the interval. But the algorithm which we get this way ....

P. Erdos, On a Problem of Sidon in Additive Number Theory, Acta Sci. Math. (Szeged), 15 (1953-54), 255-259.

....Proposition 5.1. f 2 (n)# n O(n ) Proof. A sequence of integers, a 1 , a k , forms a Sidon sequence if all the ( 2 ) sums of the form a i a j (where 1#i# j#k) are distinct. Let b 2 (n) denote the size of the largest Sidon subsequence of [n] An old Theorem of Erdo# s and Tura# n [16] states, that b 2 (n)t n. The lower bound in this theorem is supplied by Singer s Theorem [22] which states: For every prime power p there exists a sequence of integers a 1 , a 2 , a p 1 , such that the ( p 1) p differences a i a j (i j) produce all the numbers 1, 2, p( p 1) modulo ....

P. Erdo# s and P. Tura# n, On a problem of Sidon in additive number theory, and some related problems, J. London Math. Soc. 16 (

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P. Erdos, On a problem of Sidon in additive number theory, Acta Sci. Math. (Szeged) 15 (1953-54), 255-259.

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P. Erdos, On a problem of Sidon in additive number theory, Acta Sci. Math. (Szeged) 15 (1953-54), 255-259.

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