A Lebesgue Density Theorem for nonstandard cuts and an application to additive number theory Steven C. Leth March 2014 Leth () density theorem and additive number theory March 2014 1 / 27 In this talk I will outline some results obtained by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg. This research was supported by an “AIM SQuaRE” grant from the American Institute of Mathematics. Leth () density theorem and additive number theory March 2014 2 / 27 Nonstandard analysis, …rst developed by Abraham Robinson in the early 1960’s, makes use of the fact that there are other models besides the usual ones that satisfy all the same mathematical statements that can be made in First Order Logic. The goal is to exploit what is di¤erent in these models - the existence of a wide array of “actualized” limits including “in…nite” numbers - and also what is the same - all mathematical properties that can be expressed formally in …rst order logic. Sometimes we might make arguments that are “internal,” for example taking advantage of the fact that sums of in…nitely many terms still act like …nite sums. Other times we might use arguments that take advantage of an “external” view of our model - for example by using measures or equivalence relations that are not recognized entities inside the model. Leth () density theorem and additive number theory March 2014 3 / 27 To use the “best of both worlds” approach to maximum e¤ect we usually work in nonstandard models using a large language that includes symbols for every standard object we are interested in. Every subset of the natural numbers is named, for example. This helps us make strong use of the transfer principle: the nonstandard model is elementarily equivalent to the standard model, so every mathematical statement expressible in our large language is true in the nonstandard model i¤ it is true in the standard model. Leth () density theorem and additive number theory March 2014 4 / 27 The nonstandard counterpart of a standard set E is denoted by E . An internal set is one that can be de…ned inside the nonstandard model using standard sets and functions and other internal parameters. Equivalently, it is a subset of P (N), where P (N) is the power set of N. Elements of P ( N) that are not in P (N) are external sets. The set of “in…nite” natural numbers (i.e. the set N nN ) is an example of an external set. By the transfer principle every internal nonempty set has a least element since that statement is true of subsets of the natural numbers. Leth () density theorem and additive number theory March 2014 5 / 27 The nonstandard natural numbers contain the standard natural numbers as an initial segment. The “in…nite” numbers beyond those can be grouped into eqivalence classes in a number of useful ways, the most elementary being to identify two numbers as equivalent i¤ their di¤erence is …nite. These equivalence classes look like the integers, (although only using the operations of adding or subtracting …nite amounts - adding or multiplying two numbers in the same class takes us out of that equivalence class). Another simple way to classify the in…nite elements is to consider two numbers equivalent if their ratio is within an in…nitesimal of being 1, i.e. we let a b i¤ st ba = 1. Every …nite nonstandard real number r is within an in…nitesimal of exactly one real number, and this real number is denoted by st(r ). Leth () density theorem and additive number theory March 2014 6 / 27 Let H 2 N nN . There is a natural internal counting measure on [1, H ], obtained by dividing the internal cardinality of an internal set by H and taking the standard part, i.e. for every internal set E contained in [1, H ], the measure of E is de…ned to be µ(E ) := st( jHE j ), where st is the standard part mapping. This de…nes a …nitely additive measure on the algebra of internal subsets of [1, H ], which canonically extends to a countably additive probability measure on the σ-algebra of Loeb measurable subsets of [1, H ], and we will also write µ for this extension (or µ[1,H ] if we want to emphasize the interval of interest). Leth () density theorem and additive number theory March 2014 7 / 27 Let A be a standard subset of N . the lower (asymptotic) density of A is: inf d(A) := lim n !∞ Leth () jA \ [1, n]j ; n density theorem and additive number theory March 2014 8 / 27 the upper (asymptotic) density of A is: d(A) := lim sup n !∞ Leth () jA \ [1, n]j ; n density theorem and additive number theory March 2014 9 / 27 the (upper) Banach density of A is: jA \ (x + [1, n])j . n ! ∞ x 2N n BD(A) := lim sup Leth () density theorem and additive number theory March 2014 10 / 27 Nonstandard equivalents of these standard notions: Proposition 1 d(A) α i¤ there exists an H 2 N nN such that µ[1,H ] ( A) 2 3 d(A) α i¤ for all H 2 N nN µ[1,H ] ( A) α. α; BD(A) α i¤ there exists H 2 N nN and x 2 N such that µx +[1,H ] ( A) α. Leth () density theorem and additive number theory March 2014 11 / 27 Proof of number 2, as an example: d(A) α i¤ for any e > 0 there exists ne 2 N such that for all n > ne jA \ [1, n]j n α e. By transfer, this is true i¤ for all H 2 N nN and every standard e > 0 j A \ [1, H ]j H which is equivalent to µ[1,H ] ( A) Leth () α e, α. density theorem and additive number theory March 2014 12 / 27 We look at another standard combinatorial property that a subset of N might have: A N is piecewise syndetic i¤ there exists m 2 N such that for all k 2 N there exists z 2 N such that z + [1, k ] A + [1, m ]. Thus, there are arbitrarily long intervals that contain no gap in A of length greater than m. Leth () density theorem and additive number theory March 2014 13 / 27 In 2000 Jin used nonstandard methods to show that if A and B are two subsets of N with positive upper Banach densities, then A + B must be piecewise syndetic [9]. Jin and Keisler extended the result to abelian groups with tiling structures [11]. Beiglböck, M., Bergelson, V., and Fish showed that a somewhat stronger result holds in any countable amenable group. Jin’s theorem was generalized to arbitrary amenable groups by Di Nasso and Lupini [5]. At the same time, several new proofs of the theorem in [9] have appeared. For example, an ultra…lter proof is obtained by Beiglböck in [1]. A more quantitative proof that includes a bound based on the densities is obtained by Di Nasso in [4] by nonstandard methods, and in [3] by elementary means. Leth () density theorem and additive number theory March 2014 14 / 27 Jin’s original result can be thought of as analogous to the classic result in real analysis that if A, B R have positive Lebesgue measure then A + B contains an interval. In fact, if we identify two points in N as equivalent i¤ their di¤erence is …nite, then a standard set A is piecewise syndetic i¤ A contains intervals of these equivalence classes. The further extensions of Jin’s result that I will discuss today come from a new Lebesgue density theorem on these equivalence classes. Leth () density theorem and additive number theory March 2014 15 / 27 Let H 2 N nN . A cut U in [1, H ] is an initial segment of [1, H ] that is closed under addition (so the cut is always an external set). A set of the form x U, where x 2 [1, H ] is called a U monad of [1, H ] . If U = N the U-monads are the equivalence classes mentioned above - two points are equivalet i¤ their di¤erence is …nite. If U is the set of all elements that are in…nitesimal to H then the equivalence classes of x consist of all points whose distance from x is in…nitesimal with respect to H. The measure space obtained by using Loeb measure on the quotient space of [1, H ] for this cut is isomorphic to Lebesgue Measure on [0, 1] via the measure-preserving mapping that sends x U to st(x /H ). Of course this means that the Lebesgue density theorem holds in this measure space. Does it hold if U = N? In the measure space induced by every cut? Leth () density theorem and additive number theory March 2014 16 / 27 It is worth noting that the Loeb measure in the usual sense does not satisfy a similar analogue of the Lebesgue density theorem. For example the set of even numbers smaller than H has relative Loeb measure 1 /2 on every in…nite interval. However, for every cut U, the measure space induced by Loeb measure on the U monads does satisfy a Lebesgue Density Theorem. Here we will focus only on the case U = N, which is what we need for the standard results, but in fact the theorem works for all cuts. The theorem and the standard results also hold in all …nite dimensions, but for clarity here we focus on the dimension 1 case. Leth () density theorem and additive number theory March 2014 17 / 27 If E is an internal subset of N and x 2 N de…ne dE (x ) = lim inf µx +[ ν,ν] ((E + Z) \ (x + [ ν, ν])) = sup µ[ ν,ν] ν >N inf H >N N< ν <H (((E x ) + Z) \ [ ν, ν]) . De…nition If dE (x ) = 1 we say that x is a point of density of E , and we write DE for the set of points of density of E . Theorem If E [1, H ] is internal then DE is Loeb measurable, and µ[1,,H ] (DE ) = µ[1,H ] (E + Z). Leth () density theorem and additive number theory March 2014 18 / 27 So, almost every point of an internal set is a point of density, and thus almost every point of a measurable set is a point of density. Everything generalizes to arbitrary cuts and any …nite dimension, although for cuts that are not countable in their co…nality or their coinitiality there are some slight modi…cations - point of density might not quite make sense for every U measurable set. The proof of the theorem is based on the proof of the Lebesgue Density Theorem given by Faure in 2002 [6]. Leth () density theorem and additive number theory March 2014 19 / 27 If E1 and E2 are internal subsets of [1, H ], and x, y 2 N are points of density of E1 and E2 , respectively, then what can we say about x + y ? In the analogous situation in the reals it is easy to see that x + y must be contained in some interval of E1 + E2 . By a very similar argument, here we know that x + y must be contained in an interval in the quotient space of equivalence classes based on the cut U = N. But then the largest gap of E1 + E2 in this interval must be …nite, i.e. there exists a …nite m such that some interval of in…nite length containing x + y is contained in E1 + E2 + m. We say that such a point is a point of syndeticity of E1 + E2 . We note that if E1 has Loeb measure α in [1, H ] and E2 contains a point of density that is in…nitesimal with respect to H then E1 + E2 will contain at least Loeb measure α points of syndeticity of E1 + E2 . Leth () density theorem and additive number theory March 2014 20 / 27 We can use the result above to quantify the “amount of syndeticity” in sumsets. We de…ne: A N is upper syndetic of level α i¤ there exists a natural number m 2 N such that for all k 2 N, d(fn 2 N : n + [ k, k ] Leth () A + [ m, m ]g) density theorem and additive number theory α; March 2014 21 / 27 This is nearly su¢ cient to prove the following result: Theorem Let A and B be subsets of Zd with the property that d(A) = α > 0 and BD(B ) > 0. Then A + B is upper syndetic of level α. There is some work needed to get the level of syndeticity to actually be α rather than simply arbitrarily close to α for some …xed …nite m. Leth () density theorem and additive number theory March 2014 22 / 27 So, the proof of the theorem for any α e is to look at some H 2 N nN .on which the Loeb measure of A = α and note that there must be a point of density of B that is in…nitesimal to H. Then the measure of the points of syndeticity of (A + B ) on [1, H ] is at least α. Then there must exist a …nite m such that the measure of the points in (A + B ) that contain in…nite intervals with no gap of length greater than m has Loeb measure at least α e. A slightly more complicated argument is needed to remove the necessity of having including the e. Leth () density theorem and additive number theory March 2014 23 / 27 What about lower density? We can analogously de…ne A N is lower syndetic of level α i¤ there exists a natural number m 2 N such that for all k 2 N, d(fn 2 N : n + [1, k ] A + [1, m ]g) α. With considerably more work we can prove the following theoerm. In this case the e cannot be removed. Leth () density theorem and additive number theory March 2014 24 / 27 Theorem Let A and B be subsets of N with the property that d(A) = α > 0 and BD(B ) > 0. Then for any e > 0, A + B is lower syndetic of level α e. Leth () density theorem and additive number theory March 2014 25 / 27 This proof is much more di¢ cult. Even though we can easily show that on every [1, H ] the Loeb measure of the points of syndeticity of (A + B ) is at least α e, the values of the …nite distance m could change. In fact, the set C constructed below has the property that almost all points in C (on any in…nite interval) are points of syndeticity of C , and d(C ) = 1/2. However, for any m dfn 2 N : n + [ 2m, 2m ] A + [0, m ]g = 0. Example Let si be the sequence 1,2,1,2,3,1,2,3,4,1,2,3,4,5..., and let C such that: on [i !, (i + 1)!), n 2 C i¤ n f0, 1, .., si N be 1g mod 2si . Thus, on [i !, (i + 1)!), C consists of blocks of length si , with the blocks alternating between being completely contained in C and not intersecting C. Leth () density theorem and additive number theory March 2014 26 / 27 Leth () density theorem and additive number theory March 2014 27 / 27 Beiglböck, M., An ultra…lter approach to Jin’s Theorem, Isreal J. Math., 185, No 1, 369 – 374 Beiglböck, M., Bergelson, V., and Fish, Sumset phenomenon in countable amenable groups, Advances in Mathematics 223 (2010), no. 2, 416-432 Di Nasso, M., An elementary proof of Jin’s Theorem with a bound. Preprint, available at arXiv:1209.5575 Di Nasso, M.,. Embeddability properties of di¤erence sets. Preprint, available at arXiv:1201.5865 Di Nasso, M., and Lupini, M., Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups. Preprint, available at arXiv:1211.4208 Faure, C, A short proof of Lebesgue’s Density Theorem, The American Mathematical Monthly 109, no. 2 (2002), 194-196 Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981 Leth () density theorem and additive number theory March 2014 27 / 27 Hindman, N., On density, translates, and pairwise sums of integers, J. Combin. Theory Ser. A, 33 (1982), no. 2, 147–157 Jin, R., Sumset phenomenon, Proc. Amer. Math. Soc., 130(3) (2002), 855 – 861 Jin, R. Introduction of Nonstandard Methods for Number Theorists, Proceedings of the CANT (2005) Conference in Honor of Mel Nathanson, INTEGERS: The Electronic Journal of Combinatorial Number Theory, vol. 8, no. 2, (2008) Jin, R. and Keisler, H. J., Abelian groups with layered tiles and the sumset phenomenon, Trans. Amer. Math. Soc., 355(1), 79–97, 2003 Jones, F., Lebesgue Integration on Euclidean Space, revised edition, Jones and Bartlett publ., Sudbury, MA, 2001 Keisler, H. J. and Leth, S., Meager sets on the hyper…nite time line, The Journal of Symbolic Logic, Vol. 56, No 1 (1991), 71 – 102,2006 Leth () density theorem and additive number theory March 2014 27 / 27 Leth, S., Applications of nonstandard models and Lebesgue measure to sequences of natural numbers, Trans. Amer. Math. Soc., 307(2), 457-468,1988 Leth, S., Some nonstandard methods in combinatorial number theory, Studia Logica, 47(3), 85-98,1988 Leth, S., Near arithmetic progressions in sparse sets, Proc. Amer. Math. Soc., 134(6), 1579-1589, Mann, H. B., A proof of the fundamental theorem on the density of sets of positive integers, Ann. Math., Vol. 43 (1942), 523–527 Leth () density theorem and additive number theory March 2014 27 / 27