منابع مشابه
Product Sets of Arithmetic Progressions
In this paper, we generalize a result of Nathanson and Tenenbaum on sum and product sets, partially answering the problem raised at the end of their paper [N-T]. More precisely, they proved that if A is a large finite set of integers such that |2A| < 3|A| − 4, then |A2| > ( |A| `n |A| ) 2 |A|2−ε. It is shown here that if |2A| < α|A|, for some fixed α < 4, then |A2| |A|2−ε. Furthermore, if α < 3...
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Given a prime p, we say that S ⊆ Z/pZ is a critical set of density d if and only if |S| = dp and S has the least number of arithmetic progressions among all other sets S having at least dp elements. In this context, an arithemtic progression is a triple of residue classes n, n+m,n+2m modulo p. Note that this includes “trivial” progressions, which are ones wherem ≡ 0 (mod p), as well as “non-tri...
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Van der Waerden’s arithmetic sequence theorem in particular, the ‘density version’ of Szemeredi is generalized to partially ordered sets in the following manner. Let w and t be fixed positive integers and E > 0. Then for every sufficiently large partially ordered set P of width at most u’, every subset S of P satisfying ] S] > E 1 PI contains a chain a, , a2, . . . . a, such that the cardinalit...
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Nonstandard methods are used to obtain results in combinatorial number theory. The main technique is to use the standard part map to translate density properties of subsets of ∗N into Lebesgue measure properties on [0, 1]. This allows us to obtain a simple condition on a standard sequence A that guarantees the existence of intervals in arithmetic progression, all of which contain elements of A ...
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ژورنال
عنوان ژورنال: Rocky Mountain Journal of Mathematics
سال: 1987
ISSN: 0035-7596
DOI: 10.1216/rmj-1987-17-3-587