Decidability of definability issues in the theory of real addition
aa r X i v : . [ m a t h . L O ] F e b A characterization of definability in the theory of realaddition
Alexis B`esUniv. Paris Est Creteil, LACL, 94000, Creteil, France [email protected]
Christian ChoffrutIRIF (UMR 8243), CNRS and Universit´e Paris 7 Denis Diderot, France
Abstract
Given a subset of X ⊆ R n we can associate with every point x ∈ R n a vector space V ofmaximal dimension with the property that for some ball centered at x , the subset X coincidesinside the ball with a union of hyperplanes parallel with V . A point is singular if V hasdimension 0.In an earlier paper we proved that a h R , + , <, Z i -definable relation X is actually definablein h R , + , <, i if and only if the number of singular points is finite and every rational sectionof X is h R , + , <, i -definable, where a rational section is a set obtained from X by fixing somecomponent to a rational value.Here we show that we can dispense with the hypothesis of X being h R , + , <, Z i -definableby assuming that the components of the singular points are rational numbers. This providesa topological characterization of first-order definability in the structure h R , + , <, i . This paper continues the line of research started in [1]. Consider the structure h R , + , <, i of theadditive ordered group of reals along with the constant 1. It is well-known that the subgroup Z of integers is not first-order-definable. Add the predicate x ∈ Z resulting in the structure h R , + , <, Z i . In [1] we prove a topological characterization of h R , + , <, i -definable relations in thefamily of h R , + , <, Z i -definable relations, and use it to derive, on the one hand, that it is decidablewhether or not a relation on the reals definable in h R , + , <, Z i can be defined in h R , + , <, i , and onthe other hand that there is no intermediate structure between h R , + , <, Z i and h R , + , <, i (sincethen the latter result was generalized by Walsberg [4] to a large class of o − minimal structures)The topological characterization of h R , + , <, i in h R , + , <, Z i , can be described as follows. Wesay that the neighborhood of a point x ∈ R n relative to a relation X ⊆ R n has strata if there existsa direction such that the intersection of all sufficiently small neighborhoods around x with X is thetrace of a union of lines parallel to the given direction. When X is h R , + , <, i -definable, all pointshave strata, except finitely many which we call singular. In [1] we give necessary and sufficientconditions for a h R , + , <, Z i -definable relation to be h R , + , <, i -definable, namely (FSP): it hasfinitely many singular points and (DS): all intersections of X with arbitrary hyperplanes parallel to n − h R , + , <, i -definable. Theseconditions were inspired by Muchnik’s characterization of definability in Presburger Arithmetic [3](see [1] for details) 1n [1] we asked whether it is possible to remove the assumption that X is h R , + , <, Z i -definablein our characterization of h R , + , <, i -definability. In the present paper we prove that the answeris positive provided an additional assumption is required: (RSP) all singular points of X haverational components.Let us explain the structure of the proof. The necessity of the two conditions (FSP) and (DS)of our characterization of h R , + , <, i in h R , + , <, Z i in [1] is trivial. The difficult part was theirsufficiency and it used very specific properties of the h R , + , <, Z i -definable relations, in particularthe fact that h R , + , <, i - and h R , + , <, Z i -definable relations are locally indistinguishible. In orderto construct a h R , + , <, i -formula for a h R , + , <, Z i -definable relation we showed two intermediateproperties, (RB): for every nonsingular point x , the set of strata at x is a subspace which can begenerated by a set of vectors with rational coefficients, and (FI): there are finitely many ”neigh-borhood types” (i.e., the equivalence relation x ∼ y on R n which holds if there exists r > x + w ∈ X ↔ y + w ∈ X for every | w | < r ) has finite index). For general relations, the sameintermediate properties are used but (RB) and (FI) are far from being obvious and are actuallyinsufficient since we need the extra condition (RSP).We give a short outline of our paper. Section 2 gathers basic notations and definitions. In Sec-tion 3 we recall the main useful definitions and results from [1]. In Section 4 we show how the con-dition “ X is h R , + , <, Z i -definable” can be replaced by the conjunction of conditions (RSP),(RB)and (FI), then state and prove the main result. We also provide an alternative formulation of thisresult in terms of generalized projections of X . Throughout this work we assume the vector space R n is provided with the metric L ∞ (i.e., | x | =max ≤ i ≤ n | x i | ). The open ball centered at x ∈ R n and of radius r > B ( x, r ). Given x, y ∈ R n we denote [ x, y ] (resp. ( x, y )) the closed segment (resp. open segment) with extremities x, y . We use also notations such as [ x, y ) or ( x, y ] for half-open segments.Let us specify our logical conventions and notations. We work within first-order predicatecalculus with equality. We confuse formal symbols and their interpretations.We are mainly concerned with the structures h R , + , <, i and h R , + , <, Z i . Given a structure M with domain D and X ⊆ D n , we say that X is definable in M , or M -definable, if there existsa formula ϕ ( x , . . . , x n ) in the signature of M such that ϕ ( a , . . . , a n ) holds in M if and only if( a , . . . , a n ) ∈ X (this corresponds to the usual notion of definability without parameters).The h R , + , <, i -theory admits quantifier elimination in the following sense, which can be in-terpreted geometrically as saying that a h R , + , <, i -definable relation is a finite union of closedand open polyhedra. Theorem 1. [2, Thm 1] Every formula in h R , + , <, i is equivalent to a Boolean combination ofinequalities between linear combinations of variables with coefficients in Z (or, equivalently, in Q ). Most of the notions and results in this section are taken from [1]. We only give formal proofs forthe new results.
The following clearly defines an equivalence relation.2 efinition 2.
Given x, y ∈ R n we write x ∼ X y or simply x ∼ y when X is understood, if thereexists a real r > w w + y − x is a one-to-one mapping from B ( x, r ) ∩ X onto B ( y, r ) ∩ X . Example . Consider a closed subset of the plane delimited by a square. There are 10 equivalenceclasses: the set of points interior to the square, the set of points interior to its complement, thefour vertices and the four open edges.Let C l ( x ) denote the ∼ -equivalence class of x . Definition 4.
1. Given a non-zero vector v ∈ R n and a point y ∈ R n we denote L v ( y ) the linepassing through y in the direction v . More generally, if X ⊆ R n we denote L v ( X ) the set S x ∈ X L v ( x ).2. A non-zero vector v ∈ R n is an X -stratum at x (or simply a stratum when X is understood)if there exists a real r > B ( x, r ) ∩ X = B ( x, r ) ∩ L v ( X ) (1)This can be seen as saying that inside the ball B ( x, r ), the relation X is a union of linesparallel to v . By convention the zero vector is also considered as a stratum.3. The set of X -strata at x is denoted by Str X ( x ) or simply Str( x ). Proposition 5. [1, Proposition 9] For all X ⊆ R n and x ∈ R n the set Str ( x ) is a vector subspaceof R n . Definition 6.
Given a relation X ⊆ R n , the dimension dim( x ) of a point x ∈ R n is the dimensionof the subspace Str( x ). We say that x is a d -point if d = dim( x ). Moreover if d = 0 then x is saidto be X -singular, or simply singular, and otherwise it is nonsingular. Example . (Example 3 continued) Let x ∈ R . If x belongs to the interior of the square orof its complement, then Str( x ) = R . If x is one of the four vertices of the square then we haveStr( x ) = { } , i.e x is singular. Finally, if x belongs to an open edge of the square but is not a vertex,then Str( x ) has dimension 1, and two points of opposite edges have the same one-dimensionalsubspace, while two points of adjacent edges have different one-dimensional subspaces.It can be shown that all strata at x can be defined by a common value r in expression (1). Proposition 8. [1, Proposition 14] If Str ( x ) = { } then there exists a real r > such that forevery v ∈ Str ( x ) \ { } we have B ( x, r ) ∩ X = B ( x, r ) ∩ L v ( X ) . Definition 9. A X -safe radius (or simply a safe radius when X is understood) for x is a real r > r is safe then so are all 0 < s ≤ r . Byconvention every real is a safe radius if Str( x ) = { } . Example . (Example 3 continued) For an element x of the interior of the square or the interiorof its complement, let r be the (minimal) distance from x to the edges of the square. Then r issafe for x . If x is a vertex then Str( x ) = { } and every r > x . In all other cases r canbe chosen as the minimal distance of x to a vertex. Remark . If x ∼ y then Str( x ) = Str( y ) therefore given an ∼ -equivalence class E , we may defineStr( E ) as the set of common strata of all x ∈ E .Observe that the converse is false. Indeed consider, e.g., X = { ( x, y ) | y ≤ } ∪ { ( x, y ) | y = 1 } in R . The points (0 ,
0) and (0 ,
1) have the same subspace of strata, namely that generated by(1 , x y . 3t is possible to combine the notions of strata and of safe radius. Lemma 12. [1, Lemma 18] Let X ⊆ R n , x ∈ R n and r be a safe radius for x . Then for all y ∈ B ( x, r ) we have Str ( x ) ⊆ Str ( y ) . Along a stratum all points inside a ball of a safe radius are ∼ -equivalent. Lemma 13.
Let x be non-singular, v ∈ Str ( x ) \ { } , and r be safe for x . For every z ∈ B ( x, r ) we have L v ( z ) ∩ B ( x, r ) ⊆ C l ( z ) .Proof. Let z ′ be a point on L v ( z ) ∩ B ( x, r ) and s > B ( z, s ) , B ( z ′ , s ) areincluded in B ( x, r ). For every w ∈ B (0 , s ) we have z + w ∈ X ⇔ z ′ + w ∈ X . Example . (Example 3 continued) Consider a point x on an (open) edge of the square and a saferadius r for x . For every point y in B ( x, r ) which is not on the edge we have Str( x ) ⊂ Str( y ) = R .For all other points we have Str( x ) = Str( y ). We relativize the notion of singularity and strata to an affine subspace S ⊆ R n . The next definitionshould come as no surprise. Definition 15.
Given a subset X ⊆ R n , an affine subspace S ⊆ R n and a point x ∈ S , we saythat a vector v = { } parallel to S is an ( X, S )-stratum for the point x if for all sufficiently small r > S ∩ X ∩ B ( x, r ) = S ∩ L v ( X ) ∩ B ( x, r )The set of ( X, S )-strata of x is denoted Str ( X,S ) ( x ). We define the equivalence relation x ∼ ( X,S ) y on S as follows: x ∼ ( X,S ) y if and only if there exists a real r > x + w ∈ X ↔ y + w ∈ X for every w parallel to S and such that | w | < r . A point x ∈ S is ( X, S )-singular if it has no (
X, S )-stratum. For simplicity when S is the space R n we will still stick to the previous terminology andspeak of X -strata and X -singular points. Remark . Singularity and nonsingularity do not go through restriction to affine subspaces. E.g.,in the real plane, let X = { ( x, y ) | y < } and S = { ( x, y ) | x = 0 } . Then the origin is not X − singular but it is ( X, S ) − singular. All other elements of S admit (0 ,
1) as an (
X, S ) − stratumthus they are not ( X, S ) − singular. The opposite situation may occur. In the real plane, let X = { ( x, y ) | y < } ∪ S . Then the origin is X − singular but it is not ( X, S ) − singular. Let S be an affine subspace and x ∈ S . Let V the subspace generated by ( Str X ( x ) \ Str ( X,S ) ( x )) ∪{ } . If V = { } then Str X ( x ) = V + Str ( X,S ) ( x ) , and otherwise Str X ( x ) ⊆ Str ( X,S ) ( x ) .Proof. It is clear that if V = { } then every X -stratum of S is an ( X, S )-stratum.Now assume there exists v ∈ Str X ( x ) \ Str ( X,S ) ( x ). It suffices to prove that for all w ∈ Str ( X,S ) ( x ) we have w ∈ Str X ( x ). Let s > X, S ) − safe and X − safe for x .Let 0 < s ′ < s be such that L v ( z ) ∩ S ⊆ B ( x, s ) for every z ∈ B ( x, s ′ ). Let y , y ∈ B ( x, s ′ ) besuch that y − y and w are parallel. It suffices to prove the equivalence y ∈ X ↔ y ∈ X .Let y ′ (resp. y ′ ) denote the intersection point of L v ( y ) and S (resp. L v ( y ) and S ). We have y , y ′ ∈ B ( x, s ), v ∈ Str X ( x ), and s is X − safe for x , thus y ∈ X ↔ y ′ ∈ X . Similarly we have y ∈ X ↔ y ′ ∈ X . Now y ′ , y ′ ∈ B ( x, s ), y ′ − y ′ and w are parallel, and w ∈ Str ( X,S ) ( x ), whichimplies y ′ ∈ X ↔ y ′ ∈ X . 4 orollary 18. Let S be an affine subspace with underlying subspace V , and let x ∈ S be non-singular. If Str X ( x ) \ V is nonempty then Str ( X,S ) ( x ) = Str X ( x ) ∩ V . ∼ -relationLemma 19. Let X ⊆ R n , S be an affine subspace of dimension n − , y, z ∈ S , and v = { } be acommon X − stratum of y, z not parallel to S . If y ∼ ( X,S ) z then y ∼ X z .Proof. Assume y ∼ ( X,S ) z , and let r > X, S ) − safe both for y and z . Let 0 < r ′ < r be X − safe both for y and z . Since v is not parallel to S , there exists s > w ∈ R n with | w | < s , the intersection point of L v ( y + w ) (resp. L v ( z + w )) and S exists and belongs to B ( y, r ′ ) (resp. B ( z, r ′ )).It suffices to show that for every w ∈ R n with | w | < s we have y + w ∈ X ↔ z + w ∈ X . Let w ′ be such that y + w ′ is the intersection point of L v ( y + w ) and S .By our hypothesis on s , y + w ′ belongs to B ( y, r ′ ). Moreover r ′ is X − safe for y , v ∈ Str X ( y ), and w ′ − w is parallel to v , therefore y + w ∈ X ↔ y + w ′ ∈ X . Similarly we have z + w ∈ X ↔ z + w ′ ∈ X .Now | w ′ | < r ′ < r , thus by our assumptions on y, z and r we have y + w ′ ∈ X ↔ z + w ′ ∈ X .We consider here a particular case for S which plays a crucial role in expressing the character-isation stated in the main theorem and in our reasoning by induction in Section 4.3. Definition 20.
Given an index 0 ≤ i < n and a real c ∈ R consider the hyperplane H = R i × { c } × R n − i − The intersection X ∩ H is called a section of X . It is a rational section if c is a rational number.We define π H as the projection R i × { c } × R n − i − → R n − .The following facts are easy consequences of the above definitions: for all x, y ∈ H and v avector parallel to H we have:1. x ∼ ( X,H ) y if and only if π H ( x ) ∼ π H ( X ) π H ( y )2. v ∈ Str ( X,H ) ( x ) if and only if π H ( v ) ∈ Str π H ( X ) ( π H ( x )). In particular x is ( X, H ) − singularif and only if π H ( x ) is π H ( X ) − singular. In this section we describe the intersection of a ∼ -class E with a line parallel to some v ∈ Str( E ).It introduces the notion of compatibility of ∼ -classes. Lemma 21. [1, Lemma 34 and Corollary 36] Let X ⊆ R n , x ∈ R n , E = C l ( x ) and let v ∈ Str ( x ) \ { } . The set L v ( x ) ∩ E is a union of disjoint open segments (possibly infinite in one orboth directions) of L v ( x ) , i.e., of the form ( y − αv, y + βv ) with < α, β ≤ ∞ and y ∈ E .If α < ∞ (resp. β < ∞ ) then the point y − αv (resp. y + βv ) belongs to a ∼ -class F = E suchthat dim ( F ) < dim ( E ) , and we say that F is v -compatible (resp. ( − v ) − compatible) (or simplycompatible when v is undeerstood) with E . h R , + , <, i - in h R , + , <, Z i -definable relations We recall our previous characterization of h R , + , <, i -definable among h R , + , <, Z i -definable rela-tions. Recall that the notion of section is defined in Definition 20.5 heorem 22. [1, Theorem 37] Let n ≥ and let X ⊆ R n be h R , + , <, Z i -definable. Then X is h R , + , <, i -definable if and only if the following two conditions hold:(FSP) There exist only finitely many singular points;(DS) Every rational section of X is h R , + , <, i -definable. The necessity of condition (FSP) is proved by Proposition 27 of [1] and that of (DS) is trivialsince a rational section is the intersection of two h R , + , <, i -definable relations. The proof thatconditions (FSP) and (DS) are sufficient uses several properties of h R , + , <, Z i -definable relationswhich are listed in the form of a proposition below. Proposition 23.
Let n ≥ and X ⊆ R n be h R , + , <, Z i -definable. The following holds.(RSP) The components of the singular points are rational numbers [1, Proposition 27].(FI) The equivalence relation ∼ has finite index and thus the number of different spaces Str ( x ) isfinite when x runs over R n [1, Corollary 25].(RB) For all nonsingular points x , the vector space Str ( x ) has a rational basis in the sense thatit can be generated by a set of vectors with rational coefficients [1, Proposition 28]. Along with the two properties (FSP),(DS) of Theorem 22, condition (RSP) proves the “onlyif” direction of Theorem 24. These three properties are also instrumental in the proof of the“if” direction when, using property (DS), the induction on the dimension of the space reduces anarbitrary relation to an h R , + , <, i -relation. Here we show that we may remove the condition ” X is h R , + , <, Z i -definable”, i.e., state a resultfor arbitray relations, at the (modest) price of adding condition (RSP). Theorem 24.
Let n ≥ and X ⊆ R n . Then X is h R , + , <, i -definable if and only if it satisfiesthe three conditions (FSP), (DS), (RSP).(FSP) It has only finitely many singular points.(DS) Every rational section of X is h R , + , <, i -definable.(RSP) Every singular point has rational components. Observe that the three conditions are needed, as shown by the following relations which arenot h R , + , <, i -definable. • Consider the binary relation X = { ( x, x ) | x ∈ Z } . The singular elements of X are preciselythe elements of X , thus X satisfies (RSP) but not (FSP). It satisfies (DS) because everyrational section of X is either empty or equal to the singleton { ( x, x ) } for some x ∈ Z , thusis h R , + , <, i -definable. • The binary relation X = R × Z has no singular point thus it satisfies (FSP) and (RSP).However it does not satisfy (DS) since, e.g., the rational section { } × Z is not h R , + , <, i -definable. • The unary relation X = {√ } admits √ X is empty.6 roof. The necessity of the first two conditions is a direct consequence of Theorem 22, that of thethird condition is due to Proposition 23.The proof in the other direction is based on two claims 25 and 26 which show that (RB) and(FI) respectively are consequences of conditions (FSP), (DS) and (RSP).
Claim 25. If X satisfies conditions (FSP), (DS) and (RSP) then it satisfies condition (RB).Proof. We prove that for every non-singular point x ∈ R n , Str( x ) has a rational basis. If n = 1this follows from the fact that for every x ∈ R the set Str( x ) is either equal to { } or equal to R ,thus we assume n ≥ i ∈ { , . . . , n } let H i = { ( x , . . . , x n ) ∈ R n | x i = 0 } , and let us call rational i − hyperplane any hyperplane S of the form S = { ( x , . . . , x n ) ∈ R n | x i = c } where c ∈ Q (notethat the direction of S is H i ).Let x be a d − point with d ≥
1, i.e., a point for which V = Str( x ) has dimension d . For d = n the result is obvious. For 1 ≤ d < n we prove the result by induction on d .Case d = 1: It suffices to show that every 1 − point x has a stratum in Q n . Let v ∈ Str( x ) \ { } ,and let r > x . We can find i and two distinct rational i − hyperplanes S and S ,not parallel to v , and such that L v ( x ) intersects S (resp. S ) inside B ( x, r ), say at some point y (resp. y ). By Lemma 13 we have y ∼ x . By Corollary 18 it follows thatStr ( X,S ) ( y ) = Str X ( y ) ∩ H i = Str X ( x ) ∩ H i and the rightmost expression is reduced to { } since d = 1 and v H i . This implies that y is ( X, S ) − singular, i.e., that π S ( y ) is π S ( X ) − singular. Similarly y is ( X, S ) − singular, i.e., π S ( y ) is π S ( X ) − singular.By condition (DS) the rational sections X ∩ S (resp. X ∩ S ) are h R , + , <, i -definable, thusthe ( n − − ary relations π S ( X ) (resp. π S ( X )) are also h R , + , <, i -definable, and by point (RSP)of Proposition 23 this implies that π S ( y ) (resp. π S ( y )) has rational components. Thus the sameholds for y and y , and also for y − y , and the result follows from the fact that y − y ∈ Str X ( x ).Case 2 ≤ d < n : Let I ⊆ { , . . . , n } denote the set of indices i such that V H i . We have V ⊆ T i ∈{ ,...,n }\ I H i thus dim( V ) ≤ n − ( n − | I | ) = | I | , and it follows from our assumptiondim( V ) = d ≥ | I | ≥ V = P i ∈ I ( V ∩ H i ). It suffices to prove V ⊆ P i ∈ I ( V ∩ H i ), and this in turnamounts to prove that dim( P i ∈ I ( V ∩ H i )) = d . For every 1 ≤ i ≤ n we havedim( V + H i ) = dim( V ) + dim( H i ) − dim( V ∩ H i )Now if i ∈ I then dim( V + H i ) > dim( H i ) i.e. dim( V + H i ) = n , which leads to dim( V ∩ H i ) = d + ( n − − n = d −
1. Thus in order to prove dim( P i ∈ I ( V ∩ H i )) = d it suffices to show thatthere exist i, j ∈ I such that V ∩ H i = V ∩ H j . Assume for a contradiction that for all i, j ∈ I wehave V ∩ H i = V ∩ H j . Then for every i ∈ I we have V ∩ H i = V ∩ \ j ∈ I H j ⊆ \ j I H j ∩ \ j ∈ I H j = { } which contradicts the fact that dim( V ∩ H i ) = d − ≥ V = P i ∈ I ( V ∩ H i ), thus it suffices to prove that for every i ∈ I , V ∩ H i has arational basis. Let v ∈ V \ H i , and let r be safe for x . We can find a rational i − hyperplane S notparallel to v and such that the intersection point of S and L v ( x ), say y , belongs to B ( x, r ). ByLemma 13 (applied to z = x ) we have y ∼ x . Corollary 18 then implies7tr ( X,S ) ( y ) = Str X ( y ) ∩ H i = Str X ( x ) ∩ H i = V ∩ H i which yields Str π S ( X ) ( y ) = π S ( V ∩ H i )Now by condition (DS), X ∩ S is h R , + , <, i -definable, and π S ( X ) as well. By condition (RB) thisimplies that π S ( V ∩ H i ) has a rational basis, and this implies that V ∩ H i also has a rational basis. Claim 26. If X satisfies conditions (FSP), (DS) and (RSP) then it satisfies condition (FI).Proof. Before proving the claim we need a simple definition.
Definition 27.
A subset Z is X -isolated (or simply isolated when X is understood) if there existsa ∼ X -class E such that Z is the subset of elements x of E such that L v ( x ) ⊆ Z for all v ∈ Str( E ). Lemma 28.
Let X ⊆ R n satisfy (FSP) and (DS). We have1. let E be be a ∼ -class and Z ⊆ E be isolated.(a) if Str ( E ) = { } then Z is a finite union of points with rational components.(b) if Str ( E ) = { } then Z is a finite union of parallel affine subspaces with direction Str ( E ) each having a point with rational components2. There exist finitely many isolated subsets.Proof. By induction on n . For n = 1 if X is equal to R or to the empty set, the only isolated setis X and it obviously satisfies (1 b ). Otherwise every nonempty isolated set Z belongs to a ∼ -class E such that Str( E ) = { } , i.e is a union of singular points. Now by (FSP) and (DS) there existfinitely many such points and they have rational components, which implies (1 a ) and (2).Now let n ≥
1. Using the same argument as above, we know that all isolated sets Z such that Z belongs to a ∼ -class E with Str( E ) = { } satisfy (1 a ), and moreover there are finitely many suchsets Z . Thus in order to prove (2) it suffices to consider the case where Z = ∅ and Str( E ) = { } .In this case there exist v ∈ Str( E ) and i ∈ { , . . . , n } such that Z intersects the hyperplane H i .All elements of Z ∩ H i are ∼ X -equivalent thus they are also ∼ ( X,H i ) -equivalent. Furthermore forevery x ∈ Z ∩ H i we have Str ( X,H i ) ( x ) = Str X ( x ) ∩ H i by Corollary 18, and the fact that x ∈ Z implies that for every w ∈ Str X ( x ) ∩ H i we have L w ( x ) ⊆ Z ∩ H i . This shows that π H i ( x ) belongsto a π H i ( X ) − isolated class, hence π H i ( Z ) is included in a π H i ( X ) − isolated class, say W .Now by (DS) π H i ( X ) is h R , + , <, i -definable, thus by Theorem 22 it satisfies also (FSP) and(DS). By our induction hypothesis it follows that W can be written as W = S pj =1 W j , where eitherall W j ’s are parallel affine subspaces with direction π H i ( E ) each having one rational point (by (1 b )),or each W j is reduced to a point with rational components (by (1 a )). Every W j which intersects π H i ( Z ) satisfies W j ⊆ π H i ( Z ), which shows that π H i ( Z ) = S j ∈ J W j for some J ⊆ { , . . . , p } . Thatis, we have Z ∩ H i = S j ∈ J W ′ j where each W ′ j denotes the subset of H i such that π H i ( W ′ j ) = W j .Observe that if x is a rational point in W j then x ′ ∈ H j is also a rational point if π H i ( x ′ ) = x . Now Z = ( Z ∩ H i ) + Str( E ) thus Z = S j ∈ J ( W ′ j + Str( E )). Since the direction of each W ′ j is includedin Str( E ), this proves (1).For (2) we observe that Z is completely determined by Z ∩ H i , i.e π H i ( Z ). By our inductionhypothesis there are finitely many π H i ( X ) − isolated subsets hence finitely many possible sets W j ,and finitely many possible union of such sets. 8ow we turn to the proof of Claim 26. Lemma 28 shows that the number of ∼ -classes havinga non-empty isolated subset is finite. It thus suffices to prove that for every 0 ≤ d ≤ n thereexist finitely many d -classes whose isolated subset is empty. In particular, if the dimension ofsuch a ∼ -class E is non-zero there exists a ∼ -class F such that F is compatible with E anddim( E ) > dim( F ). Remember from Lemma 21 that this means that there exists y ∈ F and x ∈ E such that [ x, y ) ⊆ C l ( x ) where x − y ∈ Str( E ).For d = n there exist at most two d − classes, which correspond to elements in the interior of X or the interior of its complement.For 0 ≤ d < n we reason by induction on d and show that the number of d -classes compatiblewith d ′ -classes for some d ′ < d is finite. Since by induction the number of classes of dimension lessthan d is finite, this will prove the claim. For d = 0 the result follows from ( F SP ) and the factthat each 0 − class is a union of singular points.Now we assume 0 < d < n . By induction hypothesis there exist finitely many d ′ − classes for d ′ < d . Thus in order to meet a contradiction we assume that there exists a d ′ − class F which iscompatible with infinitely many d classes E j , j ∈ J . We may furthermore assume that for eachclass E j there is no integer d ′′ > d ′ and no d ′′ -class which is compatible with E j .We first consider the case d ′ = 0.Let y ∈ F . Because of condition (FSP), for some real s > y is the unique singularpoint in B ( y, s ). Moreover for every j ∈ J , F is compatible with E j , thus there exists a point x j ∈ E j such that [ x j , y ) ⊆ E j . Let HL j denote the open halfline with endpoint y and containing x j . Observe that we necessarily have HL j ∩ B ( y, s ) ⊆ C l ( x j ). Indeed, by Lemma 21 the condition HL j ∩ B ( y, s ) ( C l ( x j ) implies that there exists a point z = y + α ( x j − y ) ∈ B ( y, s ) such that α > z ) < d . Since y is the unique singular point in B ( y, s ) this implies dim( z ) > x j , z ) ⊆ C l ( x j ) the maximality condition stipulated for d ′ is violated.Thus, let z j be the point on HL j at distance s from y and let z be adherent to the set ofthese z j ’s. The point z is nonsingular since y is the unique singular point in the ball B ( y, s ). Let v ∈ Str( z ) \ { } . Consider some ℓ ∈ { , . . . , n } , some rational ℓ − hyperplane S such that z S and some t < s such that L v ( B ( z, t )) ∩ S ⊆ B ( z, s ). The ball B ( z, t ) contains infinitely many non ∼ -equivalent points, and by Lemma 19 their projections on S in the direction v are non ∼ ( X,S ) -equivalent. But by condition (DS) the relation X ∩ S is h R , + , <, i -definable, thus π S ( X ) satisfiescondition (FI) of Proposition 23, a contradiction.Now we consider the case where d ′ > y ∈ F . By definition y is adherent to all d -classes E j for j ∈ J . Choose some v ∈ Str( y ) andlet r be a safe radius for y . We can find 0 < s < r , k ∈ { , . . . , n } and some k − hyperplane S notparallel to v such that L v ( B ( y, s )) ∩ S ⊆ B ( y, r ). By definition of y , B ( y, s ) intersects infinitelymany pairwise distinct d − classes. Given two non ∼ − equivalent d -points z , z ∈ B ( y, s ), and w , w their respective projections over S along the direction v , we have w ( X,S ) w by Lemma19. This implies that there exist infinitely many ∼ ( X,S ) − classes. However by condition (DS), therelation X ∩ S is h R , + , <, i -definable, thus π S ( X ) satisfies condition (FI) of Proposition 23, acontradiction.Observe that X is equal to the union of ∼ -classes of its elements, thus by Claim 26, in orderto prove that X is h R , + , <, i -definable it suffices to prove that all ∼ X -classes are h R , + , <, i -definable.We prove that each ∼ -class E is definable from ∼ -classes F with smaller dimension, i.e. that E is definable in the expansion of h R , + , <, i obtained by adding a predicate for each such F . Weproceed by induction on the dimension d of Str( E ).If d = 0 then E is a union of singular points, and by (FSP) and (RSP) it follows that E is afinite subset of Q n thus is h R , + , <, i -definable.9ssume now 0 < d ≤ n . By Claim 25 there exists a rational basis V ( E ) = { v , . . . , v d } ofStr( E ). Let Z ⊆ E be isolated and let Z ′ = E \ Z . By Lemma 28 (1 b ), Z is a finite union ofparallel affine subspaces with direction V ( E ) each having a point with rational components, thus Z is h R , + , <, i -definable.It remains to prove that Z ′ is h R , + , <, i -definable. We use the following characterization of Z ′ . Lemma 29.
For every x ∈ R n , we have x ∈ Z ′ if and only if there exist ≤ p ≤ d and a sequenceof pairwise distinct elements x , . . . , x p ∈ R n such that x = x and1. for every ≤ k ≤ p − , [ x k , x k +1 ) is parallel to some u k +1 ∈ V ( E ) and does not intersectany ∼ -class F such that dim( F ) < dim( E )
2. if we denote F = C l ( x p ) then dim( F ) < dim( E ) . Moreover if x p = x p − + αu p with α = 0 then F is ( sgn ( α ) u p ) -compatible with E .Proof. We first prove that the conditions are sufficient. We prove by backward induction that[ x k , x k +1 ) ⊆ E for every 0 ≤ k ≤ p −
1. This will imply that x = x ∈ E , and the fact that x p − x belongs to Str( E ) and dim( F ) < dim( E ) will lead to x ∈ Z ′ . If k = p − x p − , x p ) is parallel to u p , and C l ( x p ) is ( sgn ( α ) u p )-compatible with E , thus [ x p − , x p ) intersects E . Moreover [ x p − , x p ) does not intersect any ∼ -class F such that dim( F ) < dim( E ), thus byLemma 21 we have [ x p − , x p ) ⊆ E . For 0 ≤ k < p −
1, by our induction hypothesis we have x k +1 ∈ E . Moreover [ x k , x k +1 ) does not intersect any ∼ -class F such that dim( F ) < dim( E ), thus[ x k , x k +1 ) ⊆ E by Lemma 21.We prove the necessity. By definition of Z ′ and Lemma 21 there exist v ∈ Str( E ) and y ∈ L v ( x )such that [ x, y ) ⊆ E and y E . Decompose v = α v i + · · · + α t v i p where 0 < i < · · · < i p ≤ d and α · · · α t = 0. We can assume w.l.o.g that v is chosen such that p is minimal. For 0 ≤ k < p set x k = x + α v i + · · · + α k v i k . By minimality of p , the segments [ x , x ) , . . . , [ x p − , x p − ) intersectno class of dimension less than dim( E ). Moreover by Lemma 21 there exists α = 0 such that[ x p − , x p − + αv p ) intersects no class of dimension less than dim( E ), and if we set x p = x p − + αv i p then x p belongs to a class F of dimension less than E and which is ( sgn ( α ) v i p )-compatible with E . In order to prove that Z ′ is h R , + , <, i -definable it suffices to show that we can express in h R , + , <, i the existence of a sequence x , . . . , x p ∈ R n which satisfies both conditions of Lemma29. Observe that V ( E ) is finite and each of its element is h R , + , <, i -definable, thus we can expressin h R , + , <, i the fact that a segment is parallel to some element of V ( E ). Moreover by (FI) thereexist finitely many ∼ -classes F such that dim( F ) < dim( E ), and all such classes are h R , + , <, i -definable by our induction hypothesis. This allows to express condition (1) in h R , + , <, i . For (2)we use again the fact that there are only finitely many classes F to consider and all of them are h R , + , <, i -definable. In this section we re-formulate Theorem 24 in terms of (generalized) projections of X .We extend the notion of section by allowing to fix several components. Definition 30. A generalized section of X is a relation of the form X s,a = { ( x , . . . , x n ) ∈ X | x s = a s . . . . , x s r = a s r } (2)where r >
0, ( s ) ,...,r = 1 ≤ s < · · · < s r ≤ n is an increasing sequence, and a = ( a s . . . . , a s r ) isan r − tuple of reals. When r = 0 we define X s,a = X by convention, i.e. X is a generalized section10f itself. If r > a are rationals then X s,a is called a rational generalized section of X .In the above definition, each X s,a is a subset of R n . If we forget the r fixed components x s , . . . , x s r we can see X s,a as a subset of R n − r , which will be called a generalized projection of X (resp. a rational generalized projection of X if X s,a is a rational generalized section of X ). Proposition 31.
For every n ≥ and every relation X ⊆ R n , X is h R , + , <, i -definable if andonly if every generalized rational projection of X has finitely many singular points and they haverational components.Proof. The proof goes by induction on n . The case n = 1 is obvious. Assume now n > X be h R , + , <, i -definable and let Y be a generalized rational projection of X . If Y = X then the result follows from Theorem 24. If Y is proper then Y is definable in h R , + , <, , X i thusit is also h R , + , <, i -definable, and the result follows from our induction hypothesis.Conversely assume that every generalized rational projection of X has finitely many singularpoints and they have rational components. We show that X satisfies all three conditions of Theorem24. Conditions (FSP) and (RSP) follow from our hypothesis and the fact that X is a generalizedrational projection of itself. It remains to prove condition (DS) namely that every rational sectionof X is h R , + , <, i -definable. This amounts to prove that every rational projection Z of X is h R , + , <, i -definable. Now Z is a generalized projection of X , and every generalized projection Y of Z is also a generalized projection of X , thus by our induction hypothesis Y has finitely manysingular points and they have rational components. Since Z is a proper projection of X , by ourinduction hypothesis it follows that Z is h R , + , <, i -definable. References [1] A.B`es and C.Choffrut. Theories of real addition with and without a predicate for integers.arXiv preprint, 2020, https://arxiv.org/abs/2002.04282 [2] J. Ferrante and C. Rackoff. A decision procedure for the first order theory of real additionwith order. SIAM J. on Computing, 4(1), 69–76, 1975.[3] A. A. Muchnik. The definable criterion for definability in Presburger arithmetic and itsapplications. Theor. Comput. Sci., 290(3), 1433–1444, 2003.[4] E. Walsberg, Generalizing a theorem of B`es and Choffrut. arXiv preprint, 2020, https://arxiv.org/abs/2002.10508https://arxiv.org/abs/2002.10508