ON AN EFFECTIVE VARIATION OF KRONECKER S APPROXIMATION THEOREM AVOIDING ALGEBRAIC SETS

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1 ON AN EFFECTIVE VARIATION OF KRONECKER S APPROXIMATION THEOREM AVOIDING ALGEBRAIC SETS LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN Abstract. Let Λ R n be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z R n be the zero locus of a finite collection of polynomials such that Λ Z or a finite union of proper full-rank sublattices of Λ. Let K 1 be the number field generated over K by coordinates of vectors in Λ, let L 1,..., L t be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K 1. For each ε > 0 a R n, we prove the existence of a vector x Λ \ Z of explicitly bounded sup-norm such that L i (x a i < ε for each 1 i t, where sts for the distance to the nearest integer. The bound on sup-norm of x depends on ε, as well as on Λ, K, Z heights of linear forms. This presents a generalization of Kronecker s approximation theorem, establishing an effective result on density of the image of Λ \ Z under the linear forms L 1,..., L t in the t-torus R t /Z t. 1. Introduction Let 1, θ 1,..., θ t be Q-linearly independent real numbers. The classical approximation theorem of Kronecker then states that the set of points {({nθ 1 },..., {nθ t } : n Z} is dense in the t-torus R t /Z t, where { } sts for the fractional part of a real number. This result was originally obtained by Kronecker [24] in 1884, presents a deep generalization of Dirichlet s 1842 theorem on Diophantine approximation [6]; see, for instance, [20] for a detailed exposition of these classical results. Kronecker s theorem can also be viewed as a statement on density of the image of the integer lattice under collection of linear forms in the torus R t /Z t (compare to the famous Oppenheim conjecture for quadratic forms. Specifically, if L 1,..., L t are linear forms in n variables with real coefficients b ij so that the set of numbers 1 b ij are linearly independent over Q, then for any ε > 0 a R t there exists x Z n such that (1 L i (x a i < ε 1 i t, 2010 Mathematics Subject Classification. 11H06, 11G50, 11J68, 11D99. Key words phrases. Kronecker s theorem, Diophantine approximation, heights, polynomials, lattices. Fukshansky was supported by the NSA grant H Simons Foundation grant # Moshchevitin was supported by RNF Grant No

2 2 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN where sts for the distance to the nearest integer. A nice survey of a wide variety of results related to Kronecker s theorem is given in [18]. Classical quantitative results in this direction are related to transference theorems for homogeneous inhomogeneous approximation for the system of linear forms L i (x (see [22], Chapter V of [3], [2]. In particular, these results give effective bounds for the size of the coordinates of the vector x in (1 under the assumption that there are effective lower bounds for max i L i (x in the homogeneous case. Some additional effective results can also be found in [26], [28]. The main goal of this note is somewhat different. We consider linear forms with algebraic coefficients extend the previously known versions of Kronecker s theorem in three ways: (1 allow for the approximating vector x as in the equation (1 above to come from an algebraic lattice Λ, (2 exclude vectors from a prescribed union Z of projective varieties or sublattices not containing this lattice, that is we are interested in approximation vectors x Λ \ Z, (3 we obtain effective constants everywhere in our upper bounds. Effective Diophantine avoidance results, exhibiting solutions to a given problem outside of a prescribed algebraic set can be viewed as statements on distribution of such solutions: not only do small solutions exist, they are also sufficiently well distributed so that it is not possible to cut them out by any finite union of varieties. In the recent years, such results were obtained in the general context of Siegel s lemma (also generalizing Faltings version of Siegel s lemma [8], [23], [7] in [10], [11], [12], [15], [16], [21], in the context of Cassels theorem on small zeros of quadratic forms its generalizations in [9], [5], [14], [17]. We will extend these investigations to Kronecker s theorem. To obtain effective constants in our bounds we use Liouville-type inequalities (see Remark 3.1 below for stronger noneffective inequalities of similar type, which can be derived from Schmidt s Subspace Theorem. To give precise statements of our results, we need some notation. 1. The lattice. Let n 1 be an integer, for each vector x R n define the sup-norm x := max 1 i n x i. Let K be a number field of degree d = r 1 + 2r 2 over Q, where r 1 r 2 are numbers of its real complex places, respectively, write O K for its ring of integers. Let 1 s w be integers, let M K w be an O K -module such that M K K = K s. Write D K (M for the discriminant of M. Define U K (M, a fractional O K -ideal in K, to be (2 U K (M = {α K : αm O w K}. We let Λ K (M R wd be the lattice of rank sd, which is the image of M under the stard Minkowski embedding. 2. The projective varieties. Let m 1 be an integer. For each 1 i m, let S i be a finite set of homogeneous polynomials in R[x 1,..., x wd ] Z(S i be its zero set in R wd, that is, Z(S i = {x R wd : P (x = 0 for all P S i }.

3 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 3 For the collection S := {S 1,..., S m } of finite sets of homogeneous polynomials, define m (3 Z S := Z(S i, (4 M S := m max{deg P : P S i }. We allow for the possibility that Z S = {0}, in which case we take instead M S = 1. Notice that Z S is an algebraic set, which is a union of a finite collection of projective varieties. Assume that the lattice Λ K (M is not contained in the set Z S. 3. The linear forms. Let K 1 = K(Λ K (M, i.e. K 1 is the number field generated over K by the entries of any basis matrix of the lattice Λ K (M. Let B := (b ij 1 i t,1 j wd be a t wd matrix with real algebraic entries so that 1, b 11,..., b t(wd are linearly independent over K 1, let l = [E : Q] where E = K 1 (b 11,..., b t(wd. We will also write l v = [E v : Q v ] for the local degree of E at every place v M(E. Define t linear forms in wd variables wd (5 L i (x 1,..., x wd = b ij x j R[x 1,..., x wd ] 1 i t. Our first goal here is to prove the following effective result on density of the image of the set Λ K (M \ Z S under the linear forms L 1,..., L t in the torus R t /Z t. Let h denote the usual Weil height on algebraic numbers, as well as its extension to vectors with algebraic coordinates; we recall the definition of height along with other necessary notation in Section 2. Theorem 1.1. Let a = (a 1,..., a t R t ε > 0. There exist x Λ K (M \ Z S p Z t such that L i (x a i p i < ε x a K (t, l, s ( sdm S D K (M s K+1 ( K 2 (wd 3 2 h(b ck (M, l, t ε l+1, where the exponent K = l 2 (t + 1 l the constants are sd 1 lt(l 1+sr1K+ a K (t, l, s = 2 2 (t + 1 3l 1 (t! 2l { c K (M, l, t = min h(α (K+1sd 1 h(α 1 K } : α U K (M. One special case of Theorem 1.1 is when Z S is a union of linear spaces, which means that the point x in question is in Λ K (M but outside of a union of sublattices of smaller rank than Λ K (M. What if the rank of such sublattices is equal to the rank of Λ K (M? The next theorem addresses this situation. Theorem 1.2. Let a = (a 1,..., a t R t ε > 0. Let m > 0 Γ 1,..., Γ m Λ K (M be proper sublattices of full rank respective determinants D 1,..., D m, let D = D 1 D m. Then for every α U K (M there exist x Λ K (M \ m Γ i p Z t such that L i (x a i p i < ε

4 4 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN x (b K (t, l, s, w ( h(αh(α 1 K D ε l+1 h(be α D K (M sm E α, where the exponent K = l 2 (t + 1 l, as in Theorem 1.1, the constant E α = b K (t, l, s, w = 2 lt(l 1+ K 2 +smr2 (t + 1 3l 1 (t! 2l (wd 3K 2, (6 E α (M, Γ 1,..., Γ m := 2 sr h(α sd 1 D K (M s 2 ( m D m D 1 sd. D i Remark 1.1. The bounds of Theorems can be recorded in a slightly weaker simplified form as x (det Λ K (M K+1 h(b K c K (M, l, t ε l+1 ( m K+2 D x (det Λ K (M K m+1 h(b K c K (M, l, t ε l+1, D i respectively, where the constants in the Vinogradov notation depend on the number field K the integer parameters t, l, s, m. The expression c K (M, l, t can be viewed as a certain measure of arithmetic complexity of M; in particular, if M O w K, then c K(M, l, t = 1. Here is a sketch of the proofs of Theorems We first construct a point y Λ K (M of controlled sup-norm, which is outside of Z S or m Γ i, respectively: in the first case, we use the classical Minkowski s Successive Minima Theorem a version of Alon s Combinatorial Nullstellensatz [1] (we use the convenient formulation developed in [13], while in the second we employ a recent result of Henk Thiel [21] on points of small norm in a lattice outside of a union of full-rank sublattices. We use y to construct an infinite sequence of points ny satisfying the above conditions, use an effective version of Kronecker s original theorem to obtain a value of the index n (depending on ε > 0 for which the required inequalities on values of linear forms are satisfied. In other words, our avoidance strategy is to follow the line ny until a necessary point is found. One may wish to use a similar strategy, but following a higher dimensional subspace of the ambient space in the hope of a better bound, however it is difficult to guarantee avoiding our fixed algebraic set with such strategy. A convenient effective version of Kronecker s theorem that we use is worked out in Section 3. It should be remarked that the most important feature of approximation results such as our Theorems is the exponent on ε in the bounds for x. As we show, this exponent is the same as in the corresponding bound of the effective version of Kronecker s theorem that we use. In Section 2 we introduce the necessary notation provide all the details of our setup. We derive an effective version of Kronecker s theorem in Section 3. We then prove Theorem 1.1 in Section 4 Theorem 1.2 in Section 5.

5 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 5 2. Notation setup Let the notation be as in Section 1. Here we introduce some additional notation needed for our algebraic setup. Let the number field K have discriminant D K, r 1 real embeddings σ 1,..., σ r1 of K, r 2 conjugate pairs of complex embeddings τ 1, τ 1,..., τ r2, τ r2, then d = r 1 + 2r 2. For each τ k, write R(τ k for its real part I(τ k for its imaginary part. Let us write M(K for the set of all places of K, then the archimedean places of K are in correspondence with the embeddings of K, we choose the absolute values v1,..., vr1 +r 2 so that for each a K a vk = σ k (a 1 k r 1 a vr1 +k = τ k(a = R(τ k (a 2 + I(τ k (a 2 1 k r 2, where sts for the usual absolute value on R or C, respectively. For each v M(K, we write K v for the completion of K at v, for each n 1 we define a local norm v : Kv n R by a v := max 1 j n a j v, for each a = (a 1,..., a n Kv n. Then the extended Weil height on K n is given by h(a = max{1, a v } dv/d, v M(K where d v = [K v : Q v ] is the local degree of K at v, so that v u d v = d for each u M(Q. For each integer n 1, define the stard Minkowski embedding ρ n K : Kn R nd by ρ n K(a := ( σ n 1 (a,..., σ n r 1 (a, R(τ n 1 (a, I(τ n 1 (a,..., R(τ n r 2 (a, I(τ n r 2 (a. We will now use Minkowski embedding to construct lattices from O K -modules outline some of their main properties; see [14] for further details. Let 1 s w be integers, let M K w be an O K -module such that M K K = K s. By the structure theorem for finitely generated projective modules over Dedekind domains (see, for instance [25], M = s β j y j : y j OK, w β j I j for some O K -fractional ideals I 1,..., I s in K. By Proposition 13 on p.66 of [25], the discriminant of M is then s (7 D K (M := D K N(I j 2, where N(I j is the norm of the fractional ideal I j. Let Λ K (M := ρ w K (M be an algebraic lattice of rank sd in Rwd, then a direct adaptation of Lemma 2 on p.115 of [25] implies that the determinant of Λ K (M is s (8 det(λ K (M = 2 sr2 D K (M s 2 = 2 sr 2 D K s 2 N(I j,

6 6 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN where the last identity follows by (7 above. Let x Λ K (M, then x = ρ w K (a for some a M (9 x 1 h(α 1, 2 for any α U K (M by inequality (54 of [14]. Let v M(K be an archimedean place, assume first that it corresponds to a real embedding σ j for some 1 j r 1, then a v = x. On the other h, if v corresponds to a complex embedding τ j ( wd 1/2 for some 1 j r 2, then a v j x2 wd x. Hence for each v, (10 x a v wd x. Let L 1,..., L t be the linear forms defined in (5. For each 1 i t, we define L i v = max 1 j wd b ij v, for each place v M(E, define the height of L i to be h(l i = h(b i1,..., b i(wd = max{1, L i v } lv/l. v M(E We similarly define the height of the matrix B to be h(b = h(b 11,..., b t(wd, then h(l i h(b for all 1 i t. We are now ready to proceed. 3. An effective version of Kronecker s theorem In this section we derive an effective version of Kronecker s theorem, which we then use to prove Theorems Similar to the setup in the beginning of Section 1, let 1, θ 1,..., θ t be Q-linearly independent real algebraic numbers. For each 1 j t, let f j (x Z[x] be the minimal polynomial of θ j of degree d j, f j be the maximum of absolute values of the coefficients of f j, A j be the leading coefficient of f j, so A j f j. By Lemma 3.11 of [29], 1 2 f j h(θ j dj d j + 1 f j, dj for every 1 j t. Define A to be the least common multiple of A 1,..., A t, so t t (11 A f j (2h(θ j dj. Let F = Q(θ 1,..., θ t be a number field of degree e t + 1, then e t d j. Let θ t+1,..., θ e 1 F be such that 1 = θ 0, θ 1,..., θ t, θ t+1,..., θ e 1 form a Q-basis for F. Let σ 1,..., σ e be the embeddings of F into C. We recall Liouville inequality. For any m = (m 0,..., m t, 0,..., 0 Z e, e e 1 (12 A e σ i (θ j m j 1, j=0

7 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 7 so e 1 (13 A ((t e + 1 max σ i(θ j m e 1 m 1 θ m t θ t 1. 1 i e,0 j t Now observe that so define (14 C 1 = C 1 (θ 1,..., θ t := Then for any 0 m Z t, max σ i(θ j max h(θ j dj, 1 i e,0 j t 1 j t ( e 1 t (t + 1 max h(θ j dj (2h(θ j edj. 1 j t (15 m 1 θ m t θ t C 1 1 m e+1. We will now apply a transference homogeneous-inhomogeneous argument. A transference principle of this sort was first described in Chapter V, 4 of [3]; the particular stronger result we are applying here is obtained in [2]. Let us write for y = (y 1,..., y t Z t, let M(y = t θ i y i L j (x = θ j x, 1 j t for x Z. Then (15 guarantees that for any 0 y Z t with y Y, M(y C 1 1 Y (e 1. Now applying the transference Lemma 3 of [2] to these linear forms, we have that for every a = (a 1,..., a t R t there exists x Z such that x 2 t ((t + 1! 2 C 1 Y e 1 max 1 j t L j(x a j 2 t ((t + 1! 2 Y 1. Letting Q = ( 2 t ((t + 1! 2 Y e 1, we obtain that max L j(x α j Q 1 e 1 1 j t for some 0 x Z with x 2 et ((t + 1! 2e C 1 Q. Taking ε = Q 1 e 1 yields the following effective version of Kronecker s theorem. immediately Theorem 3.1. Let 1, θ 1,..., θ t be Q-linearly independent real algebraic numbers, let e = [Q(θ 1,..., θ t : Q]. Let C 1 be given by (14 above, let ε > 0. Then for any (a 1,..., a t R t there exists q Z \ {0} such that (16 qθ j a j ε, 1 j t q 2 et ((t + 1! 2e C 1 ε e+1. In particular, if h(θ j H for all 1 j t max{e, d 1,..., d t } l, then ( q 2 lt(l 1 (t + 1 3l 1 (t! 2l H l2 (t+1 l ε l+1.

8 8 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN Remark 3.1. Stronger non-effective results can be derived as corollaries of Schmidt s Subspace Theorem. For instance, results discussed in Chapter 6, 2 of [27] together with the transference principles of Chapter V, 4 of [3] [2] imply, for any ε > 0 a R t under the assumptions of Theorem 3.1, the existence of q Z satisfying 16 such that q C (δε t δ, for any δ > 0, where the constant C (δ is non-effective. This would result in the same exponent on ε in the bounds for q in Theorems , but with non-effective constants. 4. Proof of Theorem 1.1 Here we present the proof of our first result. Since Λ K (M Z S, Λ K (M Z(S i for all 1 i m, so for each i at least one polynomial P i in S i is not identically zero on Λ K (M. Clearly for each 1 i m, Define Z(S i Z(P i := { x R wd : P i (x = 0 }. P (x = m P i (x, so that Z S Z(P deg(p M S, while Λ K (M Z(P. Indeed, Z(P is the union of hypersurfaces Z(P 1,..., Z(P m, a lattice cannot be covered by a finite union of hypersurfaces unless it is contained in one of them. We will next construct a point y Λ K (M of controlled sup-norm such that P (y 0. Let V = span R Λ K (M be the sd-dimensional subspace of R wd spanned by the lattice Λ K (M. For a positive real number µ, let us write C V (µ := {x V : x µ} for the sd-dimensional cube with side-length 2µ centered at the origin in V, so C V (µ = µc V (1. Let 0 < λ 1 λ 2 λ sd be the successive minima of Λ K (M with respect to the cube C V (1. In other words, for each 1 i sd, λ i := min {µ R >0 : dim R span R (Λ K (M C V (µ i}. Let v 1,..., v sd be a collection of linearly independent vectors in Λ K (M corresponding to these successive minima, then v i = λ i. Since the volume of sddimensional cube C V (1 is 2 sd, Minkowski s Successive Minima Theorem (see, for instance, [4] or [19] implies that where det(λ K (M (sd! sd v i det(λ K (M, 1 2 h(α 1 v 1 v sd, by (9. This means that (17 v 1 v sd 2h(α sd 1 det(λk (M.

9 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 9 Let I(M S = {0, 1, 2,..., M S } be the set of the first M S + 1 non-negative integers. For each ξ I(M S sd, define then (18 v(ξ = max 1 j wd sd v(ξ = sd ξ i v i, ξ i v ij sd ξ v sd sdm S 2h(α sd 1 det(λk (M, by (17. Assume that P (v(ξ = 0 for each ξ I(M S sd. Then Theorem 4.2 of [13] implies that P (x must be identically zero on V, which would contradict the fact that P does not vanish identically on Λ K (M. Hence there must exist some ξ I(M S sd such that P does not vanish at the corresponding y := v(ξ, y sdm S 2h(α sd 1 det(λk (M by (18. Since P (x is a homogeneous polynomial, it must be true that P (ny 0 for every n Z >0. On the other h, by our construction ny = n sd ξ i v i span Z {v 1,..., v sd } Λ K (M, so {ny} n Z>0 gives an infinite sequence of points in Λ K (M outside of Z S. For each such point, we have Let us define, for each 1 i t, L i (ny = nl i (y, 1 i t. wd (19 θ i := L i (y = b ij y j 0, since y j K 1, not all zero, b ij are K 1 -linearly independent. Notice that θ 1,..., θ t E, hence all of them are algebraic numbers of degree l. Let α U K (M. Then, by (10, for each archimedean v M(E, (20 max{1, θ i v } max{1, (wd 3 2 Li v y } (wd 3 2 max{1, y } max{1, Li v } 2 (wd 3 2 h(α y max{1, Li v }, by (9. By (18, y sdm S 2h(α sd 1 det(λk (M, hence (21 max{1, θ i v } sd(wd 3 2 MS 2h(α sd det(λk (M max{1, L i v }. Now suppose v M(E is non-archimedean. Then αy j is an algebraic integer for each 1 j wd, hence αy j v = α v y j v 1, meaning that Then (22 max{1, y 1 v,..., y wd v } max{1, α 1 v }. max{1, θ i v } max{1, L i v } max{1, y 1 v,..., y wd v } max{1, α 1 v } max{1, L i v },

10 10 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN for each non-archimedean v M(E. Taking a product over all places of E, we obtain: 1 h(θ i = max{1, θ i v } lv l = max{1, θ i v } lv l max{1, θ i v } lv v v M(E v sd(wd 3 2 MS 2h(α sd det(λk (Mh(L i v max{1, α 1 v } lv l sd(wd 3 2 MS 2h(α sd h(α 1 det(λ K (Mh(L i. Recalling that h(l i h(b for all 1 i t, we obtain (23 h(θ i 2 sd 2 sd(wd 3 2 MS h(α sd h(α 1 det(λ K (Mh(B, for each 1 i t, where the choice of α U K (M is arbitrary. We will now show that 1, θ 1,..., θ t are Q-linearly independent. Suppose not, then there exist c 0, c 1,..., c t Q, not all zero, such that t t wd c 0 = c i θ i = c i y j b ij, where not all c i y j are equal to zero. Recall that y Λ K (M, meaning that coordinates of y are in K 1, hence all c i y j are in K 1. This contradicts the assumption that 1, b 11,..., b 1(wd are linearly independent over K 1. Hence 1, θ 1,..., θ t must be linearly independent over Q. Now let a = (a 1,..., a t R t ε > 0, as in the statement of our theorem. Then, by (23 Theorem 3.1, there exists q Z p Z t such that q 2 lt(l 1 (t + 1 3l 1 (t! 2l ( 2 sd 3 l 2 sd(wd 2 MS h(α sd h(α 1 det(λ K (Mh(B 2 (t+1 l ε l+1 (24 qθ i a i p i < ε 1 i t. Letting x = qy, we see that qθ i = L i (x for each 1 i t x = q y. Combining these observations with (18, (24 (8 taking a minimum over all α U K (M finishes the proof of the theorem. 5. Proof of Theorem 1.2 Let Γ 1,..., Γ m be full-rank sublattices of Λ K (M of respective determinants D 1,..., D m. Let Ω = m Γ i, then Ω also has full rank D := D 1 D m det Ω. We write λ i for the successive minima of Λ K (M λ i (Ω for the successive minima of Ω. Theorem 1.2 of [21] implies that there exists y Λ K (M \ m Γ i such that ( y < det Λ m K(M D λ 1 (Ω sd 1 m λ 1 (Ω. D i

11 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 11 Our first goal is to make this bound more explicit in terms of the parameters of M. First notice that by Minkowski s Successive Minima Theorem, λ 1 (Ω ( sd λ i (Ω 1/sd (det Ω 1/sd D 1/sd. We also need a lower bound on λ 1 (Ω. Observe that λ 1 (Ω λ 1, while λ h(α 1 for any α U K (M, by (9 above. Putting these estimates together, we see that (25 y < 2h(α sd 1 det ΛK (M ( m D m D 1/sd D i for any α U K (M. Since y Λ K (M Λ K (M : Γ i = D i / det Λ K (M for each 1 i m, it follows that gd i (g Λ K (M : Γ i y = det Λ K (M y Γ i, for every g Z, hence ( ( gd1 D m (det Λ K (M m y = gd (det Λ K (M m y Ω, for every g Z. Therefore, it must be true that ( gd m (det Λ K (M m + 1 y Λ K (M \ Γ i, D (det Λ K (M m. for every g Z. For brevity, let us write D = From here on, the argument is largely similar to the proof of Theorem 1.1 above, but with some notable changes. For each 1 i t, let θ i be as in (19 for our choice of y Λ K (M \ m Γ i satisfying (25 as above, then L i ((gd + 1y = (gd + 1θ i 1 i t. Using (20 with (25 instead of (18, we obtain that max{1, θ i v } ( ( ( sd m (wd 3 D 2 2h(α det ΛK (M m D 1 sd 2h(α max{1, L i v } D i for all archimedean v M(E, while for the non-archimedean v M(E, max{1, θ i v } max{1, α 1 v } max{1, L i v }, as in (22. Taking the product over all places of E, we have for every 1 i t: (26 h(θ i (wd 3 2 2h(αh(α 1 h(b ( ( 2h(α sd 1 det ΛK (M ( m D m D 1 sd, D i 1, θ 1,..., θ t ( hence 1, D θ 1,..., D θ t are Q-linearly independent by the same reasoning as in the proof of Theorem 1.1. Now let a = (a 1,..., a t R t ε > 0, as in the statement of our theorem. Notice that for each 1 i t, (gd + 1θ i a i p i = g(d θ i + (θ i a i p i,

12 12 LENNY FUKSHANSKY AND NIKOLAY MOSHCHEVITIN for any integers p 1,..., p t. Then, applying Theorem 3.1 to approximate the vector (θ 1 a 1,..., θ t a t by the fractional parts of the integer multiples of the vector (D θ 1,..., D θ t, we conclude that there exists g Z p Z t such that g 2 lt(l 1 (t + 1 3l 1 (t! 2l ( l (wd 3 2 2h(αh(α 1 h(be α (M, Γ 1,..., Γ m 2 (t+1 l (27 ε l+1, where E α (M, Γ 1,..., Γ m is as in (6, g(d θ i + (θ i a i p i < ε 1 i t. Letting x = (gd + 1y, we see that (gd + 1θ i = L i (x for each 1 i t x = gd + 1 y. Combining these observations with (25, (27 (8 finishes the proof of the theorem. Acknowledgement. We are grateful for the wonderful hospitality of the Oberwolfach Research Institute for Mathematics: an important part of this work has been done during our Research in Pairs stay at the Institute. We would also like to thank the referee for the helpful remarks. References [1] N. Alon. Combinatorial nullstellensatz. Combin. Probab. Comput., 8(1-2:7 29, [2] Y. Bugeaud M. Laurent. On exponents of homogeneous inhomogeneous Diophantine approximation. Mosc. Math. J., 5(4: , 972, [3] J. W. S. Cassels. An introduction to Diophantine approximations. Cambridge Univ. Press, [4] J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer-Verlag, [5] W. K. Chan, L. Fukshansky, G. Henshaw. Small zeros of quadratic forms outside a union of varieties. Trans. Amer. Math. Soc., 366(10: , [6] L. G. P. Dirichlet. Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. S. B. Preuss. Akad. Wiss., pages 93 95, [7] B. Edixhoven. Arithmetic part of Faltings s proof. Diophantine approximation abelian varieties (Soesterberg, 1992, Lecture Notes in Math.(1566:97 110, [8] G. Faltings. Diophantine approximation on abelian varieties. Ann. of Math., 133(2: , [9] L. Fukshansky. Small zeros of quadratic forms with linear conditions. J. Number Theory, 108(1:29 43, [10] L. Fukshansky. Integral points of small height outside of a hypersurface. Monatsh. Math., 147(1:25 41, [11] L. Fukshansky. Siegel s lemma with additional conditions. J. Number Theory, 120(1:13 25, [12] L. Fukshansky. Algebraic points of small height missing a union of varieties. J. Number Theory, 130(10: , [13] L. Fukshansky. Algebraic points of small height missing a union of varieties. J. Number Theory, 130(10: , [14] L. Fukshansky G. Henshaw. Lattice point counting height bounds over number fields quaternion algebras. Online J. Anal. Comb., 8(5:20 pp., [15] E. Gaudron. Géométrie des nombres adélique et lemmes de siegel généralisés. Manuscripta Math., 130(2: , [16] E. Gaudron G. Rémond. Lemmes de Siegel d évitement. Acta Arith., 154(2: , [17] E. Gaudron G. Rémond. Espaces adéliques quadratiques. Math. Proc. Cambridge Philos. Soc., 162(2: , 2017.

13 EFFECTIVE KRONECKER S THEOREM AVOIDING ALGEBRAIC SETS 13 [18] S. M. Gonek H. L. Montgomery. Kronecker s approximation theorem. Indag. Math. (N.S., 27(2: , [19] P. M. Gruber C. G. Lekkerkerker. Geometry of Numbers. Second edition. North-Holl Mathematical Library, 37. North-Holl Publishing Co., Amsterdam, [20] G. H. Hardy E. M. Wright. An Introduction to the Theory of Numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York,, [21] M. Henk C. Thiel. Restricted successive minima. Pacific J. Math., 269(2: , [22] V. Jarnik. On linear inhomogeneous Diophantine approximations. Rozpravy II. Tridy Ceske Akad., 51(29:21 pp., [23] R. J. Kooman. Faltings s version of Siegel s lemma. Diophantine approximation abelian varieties (Soesterberg, 1992, Lecture Notes in Math.(1566:93 96, [24] L. Kronecker. Näherungsweise ganzzahlige Auflösung linearer Gleichungen. Monatsber. Königlich. Preuss. Akad. Wiss. Berlin, pages , , [25] S. Lang. Algebraic Number Theory. Springer-Verlag, [26] G. Malojovich. An effective version of Kronecker s theorem on simultaneous Diophantine approximation. Technical report, City University of Hong Kong, ufrj.br/~gregorio/papers/kron.pdf. [27] W. M. Schmidt. Diophantine Approximation. Springer-Verlag, [28] T. Vorselen. On Kronecker s theorem over the adéles. Master s thesis, Universiteit Leiden, [29] M. Waldschmidt. Diophantine approximation on linear algebraic groups. Springer-Verlag, Berlin, Department of Mathematics, 850 Columbia Avenue, Claremont McKenna College, Claremont, CA address: lenny@cmc.edu Steklov Mathematical Institute of the Russian Academy of Sciences, Gubkina 8, , Moscow, Russia address: moshchevitin@gmail.com