aa r X i v : . [ m a t h . L O ] A p r NIPPY PROOFS OF P-ADIC RESULTS OF DELON AND YAO
ERIK WALSBERG
Abstract.
Let K be an elementary extension of Q p , V be the set of finite a ∈ K , st be the standard part map K m → Q mp , and X ⊆ K m be K -definable. Delon has shown that Q mp ∩ X is Q p -definable. Yao has shownthat dim Q mp ∩ X ≤ dim X and dim st( V n ∩ X ) ≤ dim X . We give new NIP-theoretic proofs of these results and show that both inequalities hold in muchmore general settings. We also prove the analogous results for the expansion Q an p of Q p by all analytic functions Z mp → Q p . As an application we show thatif ( X k ) k ∈ N is a sequence of elements of an Q an p -definable family of subsets of Q mp which converges in the Hausdroff topology to X ⊆ Q mp then X is Q an p -definable and dim X ≤ lim sup k →∞ dim X k . Introduction
Fix a prime p , let K be an elementary extension of Q p , V be the set of a ∈ K such that val( a ) ≥ k for some k ∈ Z , and st be the standard part map V m → Q mp .Fact 1.1, a p -adic analogue of the Marker-Steinhorn theorem [18], is due to Delon [8]. Fact 1.1. If X ⊆ K m is K -definable then Q mp ∩ X is Q p -definable. Fact 1.2 follows from standard results on equicharacteristic zero Henselian valuedfields as V is a Henselian valuation ring. Fact 1.2. If X ⊆ K m is K -definable then st( V m ∩ X ) is Q p -definable. Let dim X be the dimension of a Q p -definable set X . Fact 1.3 is a result of Yao [36]. Fact 1.3.
Suppose that X ⊆ K m is K -definable. Then dim Q mp ∩ X ≤ dim X and dim st( V m ∩ X ) ≤ dim X. As Q mp ∩ X is a subset of st( V m ∩ X ) the first inequality is a corollary to the second,but we will see that they generalize in different directions. The analogue of Fact 1.3for o-minimal expansions of ( R , + , × ) were previously proven by van den Dries [31].We show that Fact 1.3 and its o-minimal analogue follow easily from the theoryof externally definable sets in NIP structures. The first inequality generalizes toan arbitrary elementary extension of an arbitrary NIP structure and the secondinequality generalizes to any “tame extension” of dp-minimal valued fields. Wealso show that Fact 1.1 follows from general NIP results and Fact 1.2. In Section 8we prove the analogues of Facts 1.1 and 1.3 for Q an p . In Section 9 we use theseresults to study Hausdorff limits of Q an p -definable sets, this is the p -adic analogueof o-minimal work of van den Dries [32].1.1. Acknowledgements.
Thanks to Raf Cluckers and Silvian Rideau for provid-ing a reference for Fact 8.2.
Date : April 29, 2020. Conventions and notation
By “definable” we mean “first order definable, possibly with parameters”. Through-out m, n are natural numbers, i, j, k, l are integers, and M is an L -structure. If X is an M -definable set and M ≺ N then X ( N ) is the N -definable set defined by thesame formula as X . Two structures on the same domain are interdefinable ifthey define the same sets. If x = ( x , . . . , x n ) is a tuple of variables then | x | = n .The structure induced on A ⊆ M m by M is the structure with domain A and an n -ary relation for A n ∩ X for each M -definable X ⊆ M nm . Our reference on NIPand dp-rank is Simon’s book [27]. 3. Dp rank
Definition.
To generalize Fact 1.3 we need a general notion of dimension.We use dp-rank. The dp-rank of a definable set is either a cardinal or the formalsymbol ∞ which is by declared to be larger than all cardinals. Fix an M -definableset X ⊆ M | y | . Let λ be a cardinal. An ( M , X, λ )-array consists of a sequence( ϕ α ( x α ; y ) : α < λ ) of L -formulas and an array ( a α,i ∈ M | x α | : α < λ, i < ω ) suchthat for every f : λ → ω there is b ∈ X such that M | = ϕ α ( a α,i ; b ) if and only if f ( α ) = i for all α, i. Then dim X ≥ λ if there is M ≺ N and an ( N , X ( N ) , λ )-array. If dim X ≥ λ forall cardinals λ then dim X := ∞ , we let dim X := max { λ : dim X ≥ λ } when thismaximum exists and otherwise declaredim X := sup { λ : dim X ≥ λ } − . The dp-rank of M is defined to be dim M . Of course these definitions raise thequestion of what exactly κ − κ is an infinite cardinal. There are severaloptions, and it does not matter which we select. When the structure may not beclear from context we let dim M X be the dp-rank of an M -definable set X .We will want to avoid passing to an elementary extension, so we use finitary arrays.Let Φ be a sequence ( ϕ α ( x α ; y ) : α < λ ) of L -formulas, F ⊆ λ be finite, and n ∈ N . An ( M , X, Φ , F, n )-array is an array ( a α,i ∈ M | x α | : α ∈ F, i ≤ n ) suchthat for every f : F → n there is b ∈ X such that for all α ∈ F, i ≤ n we have M | = ϕ α ( a α,i ; b ) if and only if f ( α ) = i . So dim X ≥ λ if and only if there is sucha Φ so that for every finite F ⊆ λ and n there is an ( M , X, Φ , F, n )-array.3.2. Properties.
Dp-rank characterizes NIP structures, see [27].
Fact 3.1.
Let T be a complete theory and M | = T . The following are equivalent.(1) M is NIP ,(2) dim M < ∞ ,(3) dim M < | T | + ,(4) dim X < | T | + for all M -definable sets X . Fact 3.2 shows that dp-rank is a reasonable notion of dimension. The first twoitems follow easily from the definition and the fourth is proven in [12].
Fact 3.2.
Suppose M is NIP , X, Y are definable sets, and f : X → M m is definable.(1) dim X = 0 if and only if X is finite.(2) dim X ∪ Y = max { dim X, dim Y } . (so X ⊆ Y implies dim X ≤ dim Y ), (3) dim f ( X ) ≤ dim X ,(4) If dim f − ( a ) ≤ λ for all a ∈ f ( X ) then dim X ≤ dim f ( X ) + λ . We say that M is dp-minimal when dim M ≤
1. O-minimal structures and Q p areboth dp-minimal [10]. Dp-rank is the canonical notion of dimension for definablesets in dp-minimal expansions of valued fields or divisible ordered abelian groups.Fact 3.3 is proven in [28]. Let Y be a topological space and equip Y n with theproduct topology. The naive dimension of a nonempty X ⊆ Y n is the maximal0 ≤ k ≤ n such that π ( X ) has interior for some coordinate projection π : Y n → Y k .Acl-dimension is defined in the same way as dimension is usually defined in ageometric structure (this definition makes sense in any structure). Fact 3.3.
Let M be a dp-minimal expansion of a valued field or a divisible orderedabelian group and X ⊆ M n be definable and nonempty. The following are equal:(1) The dp-rank of X ,(2) The acl -dimension of X , and(3) The naive dimension of X . So in particular dp-rank agrees with the canonical dimension for definable sets ina p -adically closed field or an o-minimal expansion of an ordered abelian group. Itfollows from Fact 3.3 that in this setting the dp-rank of X depends only on X andthe topology, not on M . We will also apply Fact 3.4, proven in [28]. Fact 3.4.
Suppose that K is a dp-minimal expansion of a valued field. Then every K -definable set is a boolean combination of closed K -definable sets. Externally definable sets
Throughout this section M ≺ N , N is highly saturated, and X ⊆ M n . We say that X is externally definable if X = M n ∩ Y for some N -definable Y . By saturationthe collection of externally definable sets does not depend on choice of N . We saythat M is Shelah complete if every externally definable set is definable. The
Shelah completion M Sh of M is the structure induced on M by N . We say that Y ⊆ N n is an honest definition of X if Y is N -definable, M n ∩ Y = X , andwhenever Z ⊆ M n is M -definable such that Z ∩ X = ∅ then Z ( N ) ∩ Y = ∅ . Thesecond claim of Fact 4.1 is a theorem of Shelah [26]. The first is due to Chernikovand Simon [3]. The second claim is a corollary to the first. Fact 4.1.
Suppose M is NIP . Every externally definable subset has an honestdefinition. Every M Sh -definable set is externally definable. It follows easily from Fact 4.1 that the Shelah completion of an NIP structure is She-lah complete, this justifies our terminology. Shelah observed that Fact 4.1 impliesthe first claim of Fact 4.2. The second claim is due to Onshuus and Usvyatsov[22].
Fact 4.2. If M is NIP then M Sh is NIP . If M is dp-minimal then M Sh is dp-minimal. The first claim of Fact 4.3 is elementary. The second claim follows from the first,Fact 4.1, and saturation.
Fact 4.3.
Suppose M ≺ O . If X ⊆ O n is externally definable in O then M n ∩ X is externally definable in M . If M is NIP then the structure induced on M by O Sh is interdefinable with M Sh . ERIK WALSBERG
Lemma 4.4 is easy and left to the reader.
Lemma 4.4.
Suppose that M is NIP , M is Shelah complete, and O ≺ M . Thenevery O Sh -definable set is of the form O n ∩ X for M -definable X ⊆ M n . The first inequality
We generalize the first inequality to arbitrary NIP structures.
Proposition 5.1.
Suppose that M is NIP , M ≺ N , and X ⊆ N m is N -definable.Then dim M Sh M n ∩ X ≤ dim N X Taking X = M we get dim M = dim M Sh . So Proposition 5.1 generalizes Fact 4.2.The proof below is essentially the same as Onshuus and Usvyatsov’s proof that M Sh is dp-minimal when M is dp-minimal [22]. Proof.
Let L Sh be the language of M Sh . If N ≺ O is highly saturated then wehave M m ∩ X = M m ∩ X ( O ), so after possibly replacing N with O we supposethat N is highly saturated. Let Y := M m ∩ X and λ be a cardinal. Suppose thatdim M Sh Y ≥ λ . Let | y | = m and fix a sequence Φ := ( ϕ α ( x α ; y ) : α < λ ) of L Sh -formulas such that for every finite F ⊆ λ and n there is a ( M Sh , Y, Φ , F, n )-array.By Fact 4.1 we have for each α ≤ λ an L -formula θ α ( x α ; y ) such that M | = ϕ α ( a ; b ) if and only if N | = θ α ( a ; b ) for all a ∈ M | x α | , b ∈ M m . Fix finite F ⊆ λ and n . Let Θ be the sequence ( θ α ( x α ; y ) : α < λ ). Observe thatif A := ( a α,i ∈ M | x α | : α ∈ F, i ≤ n ) is an ( M Sh , Y, Φ , F, n )-array then A is also an( N , X, Θ , F, n )-array. So dim N X ≥ λ . (cid:3) The second inequality
We generalize the second inequality. The results of this section are easily adaptedto expansions of divisible ordered abelian groups, we leave that to the reader.Let ( K, val) be a valued field, K be an expansion of ( K, val), and K ≺ L . Let V be the set of a ∈ L such that val( a ) ≥ val( b ) for some b ∈ K . Then L is a tameextension of K if for every a ∈ V there is b ∈ K such that val( a − b ) ≥ val( a − b ′ )for all b ′ ∈ K . It is easy to see that b must be unique, so if K ≺ L is tame then welet st : L → K be the map taking each a to the unique b = st( a ) with this property.If ( K, val) is locally compact then any elementary extension is tame. In this section K is NIP and K ≺ L is tame . Let st( L ) be the structure on K with an m -ary relation defining st( V m ∩ X ) for each L -definable X ⊆ L m . Proposition 6.1. If X ⊆ L m is L -definable then dim st( L ) st( V m ∩ X ) ≤ dim L X . So in particular st( L ) is NIP and st( L ) is dp-minimal when K is dp-minimal.It is easy to see that V is a subring of L and if a ∈ L \ V then 1 /a ∈ V , so V is a valuation subring of L . The maximal ideal m of V is the set of a ∈ L suchthat val( a ) ≥ val( b ) for all b ∈ K × . Observe that { st( a ) } = ( a + m ) ∩ K for all a ∈ V , so we may identify K with V / m . It is easy to see that st : V → K is theresidue map. We describe the associated valuation. Let Γ K , Γ L be the value groupof ( K, val) , ( L, val), respectively. Let O be the convex hull of Γ K in Γ L and w be the valuation on L given by composing val with the quotient Γ L → Γ K /O . Then V is the valuation ring of w . We now prove Proposition 6.1. Proof.
By definition O is a convex subset of Γ L so O is definable in L Sh . So w isan L Sh -definable valuation and we can regard K as an imaginary sort of L Sh , thusst : V m → K m is L Sh -definable. The proposition now follows from Fact 3.2(3). (cid:3) What is not clear at the moment is how st( L ) relates to K . Proposition 6.2. st( L ) is a reduct of K Sh .Proof. Suppose Y ⊆ L m is L -definable. Let Z := Y + m m , so Z is L Sh -definable.Note that st( V m ∩ Y ) = Z ∩ K m . So st( V m ∩ Y ) is K Sh -definable by Fact 4.3. (cid:3) In general K is not a reduct of st( L ). By [35] st( L ) cannot define a subset of Q mp which is dense and co-dense in a nonempty open set, but there are NIP expansionsof Q p which define such sets. For example Mariaule [17] shows that if H is a densefinitely generated subgroup of (1 + p Z p , × ) then ( Q p , H ) is NIP. We expect that inthis case st( L ) is interdefinable with Q p but we have not carefully checked this. Proposition 6.3.
Suppose that K is dp-minimal. Then K is a reduct of st( L ) and K Sh is interdefinable with st( L Sh ) .Proof. The proof of Proposition 6.2 shows that st( L Sh ) is a reduct of K Sh . We firstshow that K Sh is a reduct of st( L Sh ). Suppose that X ⊆ K n is K Sh -definable. Weshow that X is st( L Sh )-definable. By Facts 4.2 and 3.4 we may suppose that X isclosed. Let L ≺ N be highly saturated, Z ⊆ N n be an honest definition of X , and Y := L n ∩ Z . So Y is L Sh -definable, we show that st( V n ∩ Y ) = X . As X ⊆ Y and st is the identity of K n we have X ⊆ st( V n ∩ Y ). Fix p ∈ st( V n ∩ Y ). Weshow that p ∈ X . As X is closed it suffices to fix a val-ball B ⊆ K n containing p and show that B ∩ X = ∅ . Fix q ∈ V n ∩ Y such that st( q ) = p . Then q ∈ B ( N ) so B ( N ) ∩ Z = ∅ . As Z is honest B ∩ X is nonempty.It remains to show that K is a reduct of st( K ). Suppose X ⊆ K n is K -definable.By Fact 3.4 we may suppose X is closed. Let Y be the subset of L n defined by thesame formula as X . The proceeding paragraph shows that st( V n ∩ Y ) = X . (cid:3) Delon’s Theorem
Fact 7.1 is a well-known consequence of Pas’s quantifier elimination [23].
Fact 7.1.
Let ( M, v ) be a Henselian valued field of equicharacteristic zero withresidue field R . Every ( M, v ) -definable subset every of R m is R -definable. We now give a proof of Delon’s theorem that Q p is Shelah complete. Proof.
Let Q p ≺ L be highly saturated. So L is a tame extension as Q p is locallycompact. Let w be the valuation on N with residue map st. By the observationsabove w is a coarsening of the p -adic valuation on L , so w is Henselian as a coars-ening of a Henselian valuation is always Henselian. An application of Fact 7.1shows that st( L ) is interdefinable with Q p . Slight modifications to the proof ofProposition 6.3 show that Q Sh p and st( L ) are interdefinable. (cid:3) A subfield of Q p is an elementary substructure if and only if it is algebraically closedin Q p [11, Lemma 6.2.1]. So Corollary 7.2 follows from Fact 1.1 and Lemma 4.4. ERIK WALSBERG
Corollary 7.2.
Suppose that K is a subfield of Q p which is algebraically closed in Q p (e.g. the algebraic closure of Q in Q p ). Then X ⊆ K m is K Sh -definable if andonly if X = K m ∩ Y for some Q p -definable Y ⊆ Q mp . So the Shelah completion of K is the structure induced on K by its valuation-theoretic completion. There are several similar results. If R is a real closed subfieldof ( R , + , × ) then every R Sh -definable set is of the form R m ∩ X for ( R , + , × )-definable X ⊆ R m . More generally, suppose that R is a divisible subgroup of( R , +) and R is an o-minimal expansion of ( R, <, +). Laskowski and Steinhorn [14]show that there is a unique o-minimal expansion R (cid:3) of ( R , <, +) such that R ≺ R (cid:3) .By Marker-Steinhorn [18] R (cid:3) is Shelah complete, so every R Sh -definable set is ofthe form R m ∩ X for R (cid:3) -definable X ⊆ R m . Finally, if H is a dense subgroupof ( R , +) then every ( H, + , < ) Sh -definable set is a boolean combination of ( H, +)-definable sets and sets of the form H n ∩ X for ( R , <, +)-definable X ⊆ R m , see[34]. (If H is not n -divisible then nH = X ∩ R for any ( R , <, +)-definable X ⊆ R .)8. Q an p is Shelah complete Let Q an p be the expansion of Q p by all analytic functions Z mp → Q p for all m . Thereis a well-developed theory of Q an p -definable sets beginning with Denef and van denDries [9]. It is shown in [33] that every definable unary set in every elementaryextension of Q an p is definable in the underlying field. Fact 8.1 easily follows. Fact 8.1. Q an p is dp-minimal. So in particular dp-rank agrees with the canonical dimension on Q an p -definable sets.Suppose that Q an p ≺ L is highly saturated and let L be the underlying field of L .Note that Q an p ≺ L is tame. Let val p be the p -adic valuation, V be the set of a ∈ L such that val p ( a ) ≥ k for some k , and st : V m → Q mp be the standard part map.As above V is a valuation subring of L , the associated valuation is a coarseningof val p , and st : V → Q p is the residue map. Fact 8.2 is the analytic analogueof Fact 7.1. Fact 8.2 follows easily from a theorem of Rideau [25, Theorem 3.10].(This is closely related to the work of Cluckers, Lipshitz, and Robinson on themodel theory of valued fields with analytic structure [5, 6, 7].) Fact 8.2.
A subset X of Q mp is ( L , V ) -definable if and only if it is Q an p -definable. Following the argument of Section 7, applying Fact 8.1 when necessary, and apply-ing Fact 8.2 in place of Fact 7.1 we obtain Theorem 8.3.
Theorem 8.3. If X ⊆ L m is L -definable then Q mp ∩ X is Q an p -definable. So Q an p is Shelah complete. (This was proven in unpublished work of Hrushovski,see [21, Fact 2.6].) Proposition 8.4 follows by Proposition 6.1 and Fact 8.2. Proposition 8.4. If X is an L -definable subset of L m then dim Q an p st( X ) ≤ dim L X . Proposition 6.3 and Theorem 8.3 together yield a strengthening of Fact 8.2.
Corollary 8.5.
The structure induced on Q p by L Sh is interdefinable with Q an p . A geometric application
Following work of Br¨ocker [1, 2] and van den Dries [32] in the semialgebraic and o-minimal settings, respectively, we give a geometric application of the results above.We let | , | be the usual absolute value on Q p and declare k a k := max {| a | , . . . , | a m |} for all a = ( a , . . . , a m ) ∈ Q mp . The Hausdorff distance d H ( X, X ′ ) between bounded subsets X, X ′ of Q mp is theinfimum of t ∈ R > such that for every a ∈ X there is a ′ ∈ X ′ such that k a − a ′ k < t and for every a ′ ∈ X ′ there is a ∈ X such that k a − a ′ k < t . The Hausdorff distancebetween a bounded set and its closure is always zero. If X is a family of boundedsubsets of Q mp then X ⊆ Q mp is a Hausdorff limit of X if X is compact and thereis a sequence ( X k ) k ∈ N of elements of X such that d H ( X k , X ) → k → ∞ . Theorem 9.1.
Suppose that X is an Q an p -definable family of bounded subsets of Q mp .Any Hausdorff limit of X is Q an p -definable. If ( X k ) k ∈ N is a sequence of elements of X which Hausdorff converges to X ⊆ Q mp then dim X ≤ lim sup k →∞ dim X k . Note that any compact subset of Z mp is a Hausdorff limit of a sequence of finitesets so the restriction to definable families of sets is necessary. We will need touse Fact 9.2, an immediate consequence of the equality of naive dimension and thecanonical dimension on Q an p -definable sets. Fact 9.2.
Suppose that ( X a : a ∈ Q np ) is an Q an p -definable family of subsets of Q mp .Then { a ∈ Q np : dim X a = l } is Q an p -definable for any ≤ l ≤ m . We now proceed to prove Theorem 9.1. Our proof is very similar to that in [32] sowe omit some details. We also make a nonessential use of ultrafilter convergence.
Proof.
Let Q an p ≺ L be highly saturated. Let | x | = m and φ ( x ; y ) be a formula suchthat X is ( φ ( Q mp ; b ) : b ∈ Q | y | p ). For each k fix b k ∈ Q | y | p such that X k = φ ( Q mp ; b k ).Let u be a nonprinciple ultrafilter on N . Applying saturation fix b ∈ K | y | such thattp( b k | Q p ) → tp( b | Q p ) as k → u . Let Y := φ ( L m ; b ). It is easy to see that Y ⊆ V m and X = st( Y ). So X is Q an p -definable by Fact 8.2. By Fact 9.2 we havedim L Y = lim k → u dim X k ≤ lim sup k →∞ dim X k . So by Proposition 8.4 we have dim X ≤ lim sup k →∞ dim X k . (cid:3) Our proof of Theorem 9.1 goes through over Q p . We give an attractive formulationof the first claim in this setting. For each k ≥ P k be a unary relationdefining the set of k th powers in Q p . It is a famous theorem of Macintyre [16] thatevery paremeter free formula in the language of rings is equivalent over Q p to aboolean combination of formulas of the form f = g , val p ( f ) ≤ val p ( g ), or P k ( f ) for f, g ∈ Z [ x , . . . , x m ]. Suppose X ⊆ Q mp is Q p -definable. The complexity of X is ≤ n if X may be defined using ≤ n formulas of the form f = g, val p ( f ) ≤ val p ( g ),or P k ( f ) where each k ≤ n and each f, g ∈ Q p [ x , . . . , x m ] has degree ≤ n . It iseasy to see that Theorem 9.3 follows from saturation and the fact that a Hausdorfflimit of a sequence of elements of a Q p -definable family of sets is Q p -definable. Theorem 9.3.
For every n, m there is an l such that if ( X k ) k ∈ N is a Hausdorffconverging sequence of bounded Q p -definable subsets of Q mp , each of complexity ≤ n ,then the Hausdorff limit X of ( X k ) k ∈ N is Q p -definable of complexity ≤ l . ERIK WALSBERG
Proposition 9.4 shows that Theorem 9.1 is equivalent to the fact that st( X ) is Q an p -definable when X ⊆ V m and Proposition 8.4. Proposition 9.4.
Let Q an p ≺ L be highly saturated. Suppose that φ ( x ; y ) is aformula in the language of Q an p . Fix b ∈ L | y | and suppose X := φ ( L | x | ; b ) ⊆ V | x | .Then st( X ) is a Hausdorff limit of X := ( φ ( Q | x | p ; a ) : a ∈ Q | y | p ) . Let | x | = m and | y | = n . Given subsets X, X ′ of V m and t ∈ R > we say that d H ( X, X ′ ) < t if for every a ∈ X there is a ∈ X ′ such that k a − a ′ k < t and viceversa (we do not define d H ( X, X ′ ) in this case). Proof.
By saturation st( X ) is compact so it suffices to show that for every t ∈ R > there is a bounded Y ∈ X such that d H (st( X ) , Y ) ≤ t . Fix t ∈ R > . As X ⊆ V m it is easy to see there is a finite A ⊆ Q mp such that d H ( A, X ) < t/
2, observe that d H ( A, st( X )) ≤ t/
2. As Q an p is an elementary submodel of K we obtain a ∈ Q np such that d H ( φ ( Q mp ; a ) , A ) < t/
2. Let Y := φ ( Q mp ; a ). Note that Y is bounded.The triangle inequality for d H yields d H (st( X ) , Y ) ≤ t . (cid:3) It should be possible to give a geometric proof of Theorem 9.1 and thereby obtain ageometric proof of Fact 8.2. We are aware of two geometric proofs of the o-minimalanalogue of Theorem 9.1, Lion and Speissegger [15] and Kocel-Cynk, Pawlucki, andValette [13]. The main tool of [13] is the o-minimal Lipschitz cell decomposition,see [24], and there is now a Lipschitz cell decomposition for Q an p -definable sets [4].We believe that there is a purely geometric proof of Shelah completeness for Q p and Q an p along these lines, but we have not seriously pursued this.10. A question
Fix an o-minimal expansion R of ( R , + , × ) such that the function R > → R > given by t t r is only definable when r ∈ Q , e.g. R an . Fix λ ∈ R > and let λ Z := { λ m : m ∈ Z } . Following [30] Miller and Speissegger show that ( R , λ Z ) istame [19, Section 8.6]. It follows by [29, Theorem 4.1.2, Corollary 4.1.7] and [3,Corollary 2.6] that ( R , λ Z ) is NIP. There is a canonical notion of dimension d for( R , λ Z )-definable sets which agrees with naive, topological, and Assouad dimension.Let ( R , λ Z ) ≺ N , V be the convex hull of R in H , st be the standard part map V m → R m , and X ⊆ N m be N -definable. We believe ( R , λ Z ) is Shelah completebut we have not carefully checked this. Assuming that this is true, it is possibleto give a geometric proof that d (st( V m ∩ X )) ≤ d ( X ). There should be a NIP-theoretic proof. More specifically there should be a combinatorical invariant I M Y of a definable set Y in an NIP structure M which satisfies at least the following:(1) I ( R ,λ Z ) agrees with d ,(2) I M Sh Y = I M Y (this should be immediate from the definition and Fact 4.1),(3) I M Y ∪ Y ′ = max { I M Y, I M Y ′ } (4) I M Y × Y ′ = I M Y + I M Y ′ ,(5) I M f ( Y ) ≤ I M Y for any M -definable function f : Y → M n .If I satisfies (1) − (5) and ( R , λ Z ) is Shelah complete then we have d (st( V m ∩ X )) = I ( R ,λ Z ) st( V m ∩ X ) ≤ I N Sh X = I N X = d ( X ) . Dp-rank does not satisfy (1). Tychonievich [29] has shown that every countable( R , λ Z )-definable set is internal to λ Z and the induced structure on λ Z is inter-definable with ( λ Z , × , < ). So the dp-rank agrees with the canonical Presburgerdimension on countable ( R , λ Z )-definable sets. Furthermore the dp-rank of any un-countable ( R , λ Z )-definable set is ℵ −
1. So in this setting dp-rank is not very useful.Suppose Z ⊆ R m is ( R , λ Z )-definable. If d ( Z ) = 0 then Z is internal to λ Z andif d ( Z ) > Z → R . As ( Z , + , < ) does notinterpret an infinite field (see [35]) we have d ( Z ) > Z does not interpret an infinite field. We should have d ( Z ) = 0 if andonly if Z is “modular”. So I M X should be the “non-modular dimension” of X .We give two other examples of NIP structures to which this should apply. Thefirst example is ( S , { λ, λ λ , λ λ λ , . . . } ) where λ > S is an o-minimal expansionof ( R , + , × ) such that every S -definable function R → R is eventually boundedabove by some compositional iterate of the exponential (all known o-minimal ex-pansions of ( R , + , × ) satisfy this condition). Miller and Tyne [20] show that thisstructure is tame, in particular naive dimension is well behaved. The inducedstructure on D := { λ, λ λ , λ λ λ , . . . } should be interdefinable with ( D, < ) and anyzero-dimensional definable set should be internal to D . So I ( S ,D ) should agree withnaive dimension. Second, let Log be the Iwasawa logarithm Q × p → Q p . Mari-aule [17] shows that ( Q p , Log) is NIP. We have Log( a ) = 0 if and only if a = bp k for some k and root of unity b ∈ Q p . It follows that p Z is ( Q p , Log)-definable. It isalso shown in [17] that the induced structure on p Z is interdefinable with ( p Z , × , ⊳ )where p k ⊳ p l if and only if k < l . It should follow from [17] that naive dimensionis well behaved in ( Q p , Log) and a zero-dimensional definable set should be internalto p Z . So we expect I ( Q p , Log) to coincide with naive dimension.The vague question here is: What is the right combinatorial definition of the canon-ical dimension in structures such as ( R , λ Z ), ( S , { λ, λ λ , λ λ λ , . . . } ), or ( Q p , Log)?
References [1] L. Br¨ocker. Families of semialgebraic sets and limits. In
Real algebraic geometry (Rennes,1991) , volume 1524 of
Lecture Notes in Math. , pages 145–162. Springer, Berlin, 1992.[2] L. Br¨ocker. On the reduction of semialgebraic sets by real valuations. In
Recent advances inreal algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA,1991) , volume 155 of
Contemp. Math. , pages 75–95. Amer. Math. Soc., Providence, RI, 1994.[3] A. Chernikov and P. Simon. Externally definable sets and dependent pairs.
Israel J. Math. ,194(1):409–425, 2013.[4] R. Cluckers, G. Comte, and F. Loeser. Lipschitz continuity properties for p -adic semi-algebraicand subanalytic functions. Geom. Funct. Anal. , 20(1):68–87, 2010.[5] R. Cluckers and L. Lipshitz. Fields with analytic structure.
J. Eur. Math. Soc. (JEMS) ,13(4):1147–1223, 2011.[6] R. Cluckers and L. Lipshitz. Strictly convergent analytic structures.
J. Eur. Math. Soc.(JEMS) , 19(1):107–149, 2017.[7] R. Cluckers, L. Lipshitz, and Z. Robinson. Analytic cell decomposition and analytic motivicintegration.
Ann. Sci. ´Ecole Norm. Sup. (4) , 39(4):535–568, 2006.[8] F. Delon. D´efinissabilit´e avec param`etres ext´erieurs dans Q p et R . Proc. Amer. Math. Soc. ,106(1):193–198, 1989.[9] J. Denef and L. van den Dries. p-adic and real subanalytic sets.
The Annals of Mathematics ,128(1):79, July 1988. [10] A. Dolich, J. Goodrick, and D. Lippel. Dp-minimality: Basic facts and examples.
Notre DameJournal of Formal Logic , 52(3):267–288, July 2011.[11] A. J. Engler and A. Prestel.
Valued fields . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.[12] I. Kaplan, A. Onshuus, and A. Usvyatsov. Additivity of the dp-rank.
Trans. Amer. Math.Soc. , 365(11):5783–5804, 2013.[13] B. Kocel-Cynk, W. a. Pawlucki, and A. Valette. A short geometric proof that Hausdorff limitsare definable in any o-minimal structure.
Adv. Geom. , 14(1):49–58, 2014.[14] M. C. Laskowski and C. Steinhorn. On o-minimal expansions of Archimedean ordered groups.
J. Symbolic Logic , 60(3):817–831, 1995.[15] J.-M. Lion and P. Speissegger. A geometric proof of the definability of Hausdorff limits.
Selecta Math. (N.S.) , 10(3):377–390, 2004.[16] A. MacIntyre. On definable subsets of p-adic fields.
The Journal of Symbolic Logic , 41(3):605,Sept. 1976.[17] N. Mariaule. Model theory of the field of p -adic numbers expanded by a multiplicative sub-group. arXiv:1803.10564 , 2018.[18] D. Marker and C. I. Steinhorn. Definable types in o-minimal theories. J. Symbolic Logic ,59(1):185–198, 1994.[19] C. Miller. Tameness in expansions of the real field. In
Logic Colloquium ’01 , volume 20 of
Lect. Notes Log. , pages 281–316. Assoc. Symbol. Logic, Urbana, IL, 2005.[20] C. Miller and J. Tyne. Expansions of o-minimal structures by iteration sequences.
NotreDame J. Formal Logic , 47(1):93–99, 2006.[21] A. Onshuus and A. Pillay. Definable groups and compact p -adic lie groups.
Journal of theLondon Mathematical Society , 78(1):233–247, May 2008.[22] A. Onshuus and A. Usvyatsov. On dp-minimality, strong dependence and weight.
J. SymbolicLogic , 76(3):737–758, 2011.[23] J. Pas. Uniform p-adic cell decomposition and local zeta functions.
Journal fr die reine undangewandte Mathematik , 399:137–172, 1989.[24] W. a. Paw l ucki. Lipschitz cell decomposition in o-minimal structures. I.
Illinois J. Math. ,52(3):1045–1063, 2008.[25] S. Rideau. Some properties of analytic difference valued fields.
J. Inst. Math. Jussieu ,16(3):447–499, 2017.[26] S. Shelah. Dependent first order theories, continued.
Israel J. Math. , 173:1–60, 2009.[27] P. Simon.
A guide to NIP theories , volume 44 of
Lecture Notes in Logic . Cambridge UniversityPress, 2015.[28] P. Simon and E. Walsberg. Tame topology over dp-minimal structures.
Notre Dame J. Form.Log. , 60(1):61–76, 2019.[29] M. A. Tychonievich.
Tameness results for expansions of the real field by groups . ProQuestLLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–The Ohio State University.[30] L. van den Dries. The field of reals with a predicate for the powers of two.
Manuscripta Math. ,54(1-2):187–195, 1985.[31] L. van den Dries. T-convexity and tame extensions II.
Journal of Symbolic Logic , 62(1):14–34,Mar. 1997.[32] L. van den Dries. Limit sets in o-minimal structures. In
O-minimal Structures, Proceedingsof the RAAG Summer School Lisbon 2003, Lecture Notes in Real Algebraic and AnalyticGeometry. Cuvillier . Verlag, 2005.[33] L. van den Dries, D. Haskell, and D. Macpherson. One-dimensional p -adic subanalytic sets. J. London Math. Soc. (2) , 59(1):1–20, 1999.[34] E. Walsberg. Dp-minimal expansions of ( Z , +) via dense pairs via mordell-lang. forthcoming .[35] E. Walsberg. Externally definable quotients and nip expansions of the real ordered additivegroup, 2019, arXiv:1910.10572.[36] N. Yao. On dimensions, standard part maps, and p -adically closed fields. arXiv:2002.10117 ,2020. Department of Mathematics, Statistics, and Computer Science, Department of Math-ematics, University of California, Irvine, 340 Rowland Hall (Bldg.
E-mail address : [email protected] URL ::