Multiplicative structures in additively large sets

Previous research extending over a few decades has established that multiplicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an infinite sequence must contain all of the initial products from another infinite sequence. And, as a corollary of a result of Renling Jin, we show that if A and B have positive upper Banach density, then A + B contains all of the initial products from an infinite sequence. We also show that if a set has a complement which is not additively piecewise syndetic, then any translate of that set is both additively and multiplicatively large in several senses.We investigate whether a subset of N with bounded gaps--a syndetic set--must contain arbitrarily long geometric progressions. We believe that we establish that this is a significant open question.

[1]  E. Szemerédi.  On sets of integers containing k elements in arithmetic progression , 1975 .

[2]  J. Spencer.  Ramsey Theory , 1990 .

[3]  Paul Erdös,et al.  On sequences of positive integers , 1936 .

[4]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[5]  Neil Hindman.  Partitions and sums and products of integers , 1979 .

[6]  Daniel M. Gordon,et al.  On sequences without geometric progressions , 1996, Math. Comput..

[7]  R. A. Rankin,et al.  XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression , 1961, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[8]  E. Haacke.  Sequences , 2005 .

[9]  Neil Hindman,et al.  Ultrafilters and combinatorial number theory , 1979 .

[10]  Renling Jin.  The sumset phenomenon , 2001 .

[11]  Neil Hindman,et al.  Some new results in multiplicative and additive Ramsey theory , 2008 .

[12]  Neil Hindman,et al.  Nonmetrizable topological dynamics and Ramsey theory , 1990 .

[13]  A. Rényi.  On the density of certain sequences of integers. , 1955 .

[14]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[15]  Vitaly Bergelson,et al.  Multiplicatively large sets and ergodic Ramsey theory , 2005 .

[16]  Rudolf Ahlswede,et al.  On the quotient sequence of sequences of integers , 1999 .

[17]  SOCIETATIS,et al.  On divisibility properties of sequences of integers bY , 1968 .

[18]  Neil Hindman,et al.  Partition Regular Structures Contained in Large Sets Are Abundant , 2001, J. Comb. Theory, Ser. A.

[19]  Neil Hindman,et al.  Algebra in the Stone-Cech Compactification: Theory and Applications , 1998 .

[20]  Dona Strauss.  The smallest ideals of βN under addition and multiplication , 2005 .

[21]  Neil Hindman,et al.  Notions of Size and Combinatorial Properties of Quotient Sets in Semigroups , 1998 .

[22]  Ralph Alexander.  Density and multiplicative structure of sets of integers , 1967 .

[23]  P. Erdös.  Note on Sequences of Integers No One of Which is Divisible By Any Other , 1935 .