By Melvyn B. Nathanson

This court cases quantity relies on papers provided on the Workshops on Combinatorial and Additive quantity concept (CANT), which have been held on the Graduate heart of the town college of latest York in 2011 and 2012. The objective of the workshops is to survey contemporary growth in combinatorial quantity thought and comparable elements of arithmetic. The workshop draws researchers and scholars who talk about the cutting-edge, open difficulties and destiny demanding situations in quantity theory.

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**Sample text**

Proof. A/ D 1. jan bn j/n 1 is bounded. Proposition 1 ([6]). If A and B are close infinite subsets of N, then they are in the same, lower or upper, class. Proof. e. bi ; bj is a B-representation of an element of the interval of integers I D Œm 2d; m C 2d . B/ D 1. t u Proposition 2. Let A D fx1 Ä x2 Ä Ä xn Ä g D fxn W n 2 J g and B D fy1 Ä y2 Ä Ä yn Ä g D fyn W n 2 J g, where J N , be two (not necessarily strictly) increasing sequences in N having the same (finite or infinite) index set J , such that jxn yn j Ä d for all n 2 J , with a fixed d 2 N .

293, 595–610 (2005) 16. T. Chapman, D. Steinberg, Elasticity in generalized arithmetical congruence monoids. Results Math. 58, 221–231 (2010) 17. A. Geroldinger, F. Halter-Koch, Congruence monoids. Acta Arith. 112, 263–296 (2004) 18. A. Geroldinger, F. Halter-Koch, Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278 (Chapman & Hall/CRC, Boca Raton, 2006) 19. A. Geroldinger, F. Halter-Koch, Non-unique factorizations: a survey, in Multiplicative Ideal Theory in Commutative Algebra (Springer, New York, 2006), pp.

Thus A and B are in the same, lower or upper, class. Proof. As in the proof of Proposition 1, to every A-representation xi ; xj of an integer m; there corresponds a B-representation yi ; yj of one of the 4d C 1 integers n in the interval I D Œm 2d; m C 2d . i; j / 7! e. xi 2d Ä xk Ä xi C 2d; 46 L. Haddad and C. e. 4d . B/. A/. B/ D 1. t u < an < g be a subset of N, ˛ a positive Proposition 3. Let A D fa1 < a2 < real number, and B D fŒ˛an W n 2 N g, where Œx denotes the integral part of a real number x.