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In mathematics, the sumset of two subsets A and B of an (additive) abelian group is formed by adding each element of A to each element of B: A + B = {...

In additive combinatorics, the Erdős sumset conjecture is a conjecture which states that if a subset A {\displaystyle A} of the natural numbers N {\displaystyle...

In additive number theory and combinatorics, a restricted sumset has the form S = { a 1 + ⋯ + a n :   a 1 ∈ A 1 , … , a n ∈ A n   a n d   P ( a 1 , … ...

theory and the geometry of numbers. Principal objects of study include the sumset of two subsets A and B of elements from an abelian group G, A + B = { a...

study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about the structures of A and B? In the...

central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if | A + A | / | A | {\displaystyle |A+A|/|A|}...

Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus in 2021. The Erdős sumset conjecture on sets, proven by Joel Moreira, Florian Karl Richter and Donald...

b : a , b ∈ A } {\displaystyle A+A=\{a+b:a,b\in A\}} and is called the sumset of A {\displaystyle A} . The set of pairwise products is A ⋅ A = { a ⋅ b...

theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation...

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1...

arithmetic. There is an interesting relation between the operation of forming sumsets of subsets of nonnegative integers and lunar multiplication on binary numbers...

Plünnecke–Ruzsa inequality is an inequality that bounds the size of various sumsets of a set B {\displaystyle B} , given that there is another set A {\displaystyle...

particular, the above result implies that Lpθ is included in Lp0 + Lp1, the sumset of Lp0 and Lp1 in the space of all measurable functions. Therefore, we have...

{\displaystyle k} or fewer elements of S {\displaystyle S} . That is, the sumset of k {\displaystyle k} copies of S {\displaystyle S} consists of all natural...

of a result of Jean Bourgain of the size of arithmetic progressions in sumsets, as well as a proof of the Cameron–Erdős conjecture on sum-free sets of...

Bunkbed conjecture (Nikita Gladkov, Igor Pak, Alexander Zimin, 2025). Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018). McMullen's...

argument, all Golomb rulers must be Sidon sets. Moser–de Bruijn sequence Sumset Erdős, P.; Turán, P. (1941). "On a problem of Sidon in additive number theory...

polynomial growth. If A is a set of N integers, how large or small can the sumset A + A := { x + y : x , y ∈ A } , {\displaystyle A+A:=\{x+y:x,y\in A\},}...

B=\{0=b_{1}<b_{2}<\cdots \}} are two sequences of natural numbers. We write A + B for the sumset, that is, the set of all elements a + b where a is in A and b is in B; and...

In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p...

{\mathsf {Z}}^{-}} , axiomatized as extensionality, pair set, power set, sumset, and stratified Δ 0 P {\displaystyle \Delta _{0}^{\mathcal {P}}} separation...

Erdős–Hajnal conjecture Erdős–Gyárfás conjecture Erdős–Straus conjecture Erdős sumset conjecture Erdős–Szekeres conjecture Erdős–Turán conjecture (disambiguation)...

refer to one of several related theorems regarding the sizes of certain sumsets in abelian groups. These are named after Martin Kneser, who published them...

ME. Solymosi, József (2009), "Bounding multiplicative energy by the sumset", Advances in Mathematics, 222 (2): 402–408, arXiv:0806.1040, doi:10.1016/j...

problem sorts the sumset, the set of sums of pairs, with duplicate sums condensed to a single value. For this variant, the size of the sumset may be significantly...

{\displaystyle A} and B {\displaystyle B} are subsets of a group, then the sumset notation A + B {\displaystyle A+B} is used to denote { a + b : a ∈ A , b...

{G}}^{2})=1} . (Here the symbol A ⊕ B {\displaystyle A\oplus B} denotes the sumset of A ∪ { 0 } {\displaystyle A\cup \{0\}} and B ∪ { 0 } {\displaystyle B\cup...

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Springer. ISBN 978-0-387-94655-9. Odlyzko, A. M.; Schonhage, A. (1988)...

Taxicab number Generalized taxicab number Cabtaxi number Schnirelmann density Sumset Landau–Ramanujan constant Sierpinski number Seventeen or Bust Niven's constant...

reference discusses much of the literature on the convex hulls of Minkowski sumsets in its "Chapter 3 Minkowski addition" (pages 126–196): Schneider, Rolf...

(2007). "Lattice points on circles, squares in arithmetic progressions and sumsets of squares". In Granville, Andrew; Nathanson, Melvyn Bernard; Solymosi...

proves theorems on the density of sumsets of sequences, including Mann's theorem that the Schnirelmann density of a sumset is at least the sum of the Schnirelmann...

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively...

1214/aoms/1177703855. JSTOR 2238330. Kemperman, J. H. B. (1960). "On small sumsets in an abelian group". Acta Mathematica. 103 (1–2): 63–88. doi:10.1007/BF02546525...

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165 (1st ed.). Springer-Verlag....

Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer. ISBN 0-387-94655-1...

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. pp. 180–185...

In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures)...

proved the Schnirelmann–Landau conjecture on the asymptotic density of sumsets in 1942. By doing so he established Mann's theorem and earned the 1946...

topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic...

the geometry of numbers, these ideas have been applied to the study of sumsets, one of the key objects of additive combinatorics. Minkowski's geometry...

1.38. Ruzsa, I. Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica. 65 (4): 379–388. doi:10.1007/BF01876039....

structured sets of values, including heap-ordered trees, sorted matrices, and sumsets. Another simple example is a selection algorithm, to find the k {\displaystyle...

|}\langle \mathbf {x} ,A\mathbf {x} \rangle {\big |}.} Let sum of sets denote a sumset. General properties The numerical range is the range of the Rayleigh quotient...

ISBN 978-0-387-94656-6) Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Melvyn B. Nathanson (1996, ISBN 978-0-387-94655-9) Differential Geometry —...

Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever...

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. ISBN 978-0-387-94655-9...

2997–3002. doi:10.1090/s0002-9939-2012-11156-6. Ruzsa, Imre Z. (2009). "Sumsets and structure". Combinatorial number theory and additive group theory....

(1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer-Verlag. ISBN 0-387-94655-1...

Paul Erdős in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problems or EGZ Theorem. With Stephen Redmond, he posed the...

2012. Ruzsa, I. Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica. 65 (4): 379–388. doi:10.1007/bf01876039....