Combinatorial and Additive Number Theory: CANT 2011 and 2012 by Melvyn B. Nathanson

By Melvyn B. Nathanson

This complaints quantity relies on papers offered on the Workshops on Combinatorial and Additive quantity idea (CANT), which have been held on the Graduate middle of town college of latest York in 2011 and 2012. The aim of the workshops is to survey contemporary development in combinatorial quantity idea and similar components 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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E. A/ Ä 2: Obviously, Sidon sets are in the lower class. A is called sparse (resp. quasi-sparse) if anC1 > 2an (resp. if anC1 2an and a1 > 0/ for all indices n: Proposition 4. If A is sparse or quasi-sparse, then A is a Sidon set, and therefore A is in the lower class L. Proof. Assume that ai C aj D am C an ; with i Ä j and m Ä n: If n > j; then n j C 1; and therefore an aj C1 > 2aj when A is sparse (resp. an 2aj when A is quasi-sparse), so that am C an > am C 2aj am C ai C aj ai C aj (resp. am C an am C 2aj am C ai C aj > ai C aj ; since am a1 > 0/; contradicting the assumption.

It follows from the fact that Ci is not a convergent (by the minimality of H ), and induction applied to Ai ; Bi . A1 C H1 /j D jH j jA1 C H1 j contradicts Eq. 1 for i D 1. We get a similar contradiction under the assumption that A2 D ;. Thus A2 ; B2 ¤ ; and Eq. 1 holds for i D 1; 2. A1 [ B1 / C H1 /j jH j jA1 C H1 j jB1 C H1 j which contradicts Eq. 1. Therefore, a C H ¤ b C H . Our next inequality follows from the observation that the left-hand side of Eq. 1 is nonnegative and all terms on the right-hand side are multiples of jHi j.

2. 3. 4. 5. Ma;b / D 1, ˇ D 1, p 2 Ma;b , and a D p by minimality of a. ˛ C 1/ 1/=˛ D 2. Ma;b / < 2 implies ˇ D ˛, so that d D p ˛ 2 Ma;b . By minimality of a, we conclude a D d D p ˛ . Ma;b / < 2. 4/ is involved; the interested reader should consult [7]. t u To this point, we have seen that whenever an ACM has finite (and rational) elasticity, its elasticity is accepted. This is no longer the case for local ACMs, as the following example testifies. 10. Consider M4;6 . Then p D 2, ˛ D 1, and ˇ D 2 since clearly 4 2 M4;6 .

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