Let ⟨ M , F , μ , T ⟩ be a dynamical system with T − invariant probability measure μ.

Let A n ⊂ M be a sequence of measurable sets. We set B n : = T − n A n . Define the set lim sup B n as lim sup B n : = ⋂ N = 1 ∞ ⋃ n = N ∞ B n A classical Borel-Cantelli lemma in probability theory states:

The last lemma can be restated as follows:

Next we introduce several necessary definitions.

A sequence of subsets A n ⊂ M is called a Borel-Cantelli (BC) sequence if for μ − almost every x ∈ M there are infinitely many values of n ∈ N, such that, T n x ∈ A n .

Denote by χ n ( x ) the indicator function of the set B n : = T − n A n . For every N > 0 we define the following two sums: S N ( x ) : = ∑ n = 1 N χ n ( x ) , E N : = ∑ n = 1 N μ ( A n ) .

We define the quantities R m n which characterizes the dependence of two events B m and B n : R m n : = μ ( B m ∩ B n ) − μ ( B m ) μ ( B n ) = μ ( T − m A m ∩ T − n A n ) − μ ( A m ) μ ( A n ) .

A sufficient condition for a sequence { A n } to be an sBC sequence, in terms of R m n , was first found by W. Schmidt, and the proof was later provided by Sprindzuk [13] in the context of Diophantine approximations. This condition was recently adapted to dynamical systems by D. Kleinbock and G. Margulis [9].

We suppose that the numbers R m n satifies the following condition: ∑ n = M N ∑ m = M N | R m n | ≤ C · ∑ n = M N μ ( A n ) , for some constant C > 0 and for all N > M > 1. The last condition is called ( S P ) − condition. Theorem 1

If the sequence { A n } satisfies (SP), then it is an sBC sequence; moreover, for μ-almost every x ∈ X one has (2) S N = E N + O ( E N 1 / 2 log 3 / 2 + ε E N ) .

We now introduce cylinder sets in symbolic spaces. An alphabet A is a finite set of symbols; for example, A = { a 1 , a 2 , … , a k }.

A cylinder C ⊂ { 0 , 1 } Z is obtained by fixing symbols on a finite interval L = [ m , k ] ⊂ Z, i.e., for some ω = { ω m , … , ω k } ∈ { 0 , 1 } L , we set (3) C L ( ω ) = { x ∈ { 0 , 1 } Z : x i = ω i , i ∈ L } .

Each cylinder is open and closed in { 0 , 1 } Z . We call m and k the left and right endpoints of an interval L, respectively, and ( m + k ) / 2 the center of L. Following [4], we introduce the relevant geometry explicitly. If L = [ m , k ] ⊂ Z is a finite lattice interval and D ∈ N 0 , its D-neighborhood is N D ( L ) : = [ m − D , k + D ] ∩ Z . We say that an interval L 2 lies in the D-neighborhood of L 1 if L 2 ⊂ N D ( L 1 ). Definition 3

Two lattice intervals L 1 and L 2 are called D-nested if either L 1 ⊂ N D ( L 2 ) or L 2 ⊂ N D ( L 1 ).

Let S 1 = [ 0 , 1 ) be the unit circle and let f : S 1 → [ 0 , 1 ] be a Borel function with Fourier coefficients (4) f ˆ ( k ) = ∫ 0 1 f ( t ) e − 2 π i k t d t . Define the convolution kernel K f ( n , m ) = f ˆ ( n − m ) , n , m ∈ Z . Then K f induces a stationary determinantal measure μ f on { 0 , 1 } Z via (5) μ f ( { x ∈ { 0 , 1 } Z ∣ x n 1 = ⋯ = x n k = 1 } ) = det ( f ˆ ( n i − n j ) ) 1 ≤ i , j ≤ k . The process is shift invariant under σ ( x ) n = x n + 1 .

Suppose that a function f is given in the form (4). The n × n Toeplitz matrix generated by f is defined by T n ( f ) = ( f ˆ ( i − j ) ) 1 ≤ i , j ≤ n . Equivalently, T n ( f ) has the explicit form T n ( f ) = f ˆ ( 0 ) f ˆ ( − 1 ) f ˆ ( − 2 ) ⋯ f ˆ ( − ( n − 1 ) ) f ˆ ( 1 ) f ˆ ( 0 ) f ˆ ( − 1 ) ⋯ f ˆ ( − ( n − 2 ) ) f ˆ ( 2 ) f ˆ ( 1 ) f ˆ ( 0 ) ⋯ f ˆ ( − ( n − 3 ) ) ⋮ ⋮ ⋮ ⋱ ⋮ f ˆ ( n − 1 ) f ˆ ( n − 2 ) f ˆ ( n − 3 ) ⋯ f ˆ ( 0 ) .

We denote by 1 ∈ R n the vector whose all components are equal to 1.

For a v = ( v 1 , … , v n ) ∈ C n , we denote by D n ( v ) the diagonal matrix associated with v, defined by D n ( v ) : = diag ( v ) = v 1 0 0 ⋯ 0 0 v 2 0 ⋯ 0 0 0 v 3 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ v n .

Let ε = ( ε i ) i = 1 n ∈ { 0 , 1 } N and let T n ( f ) = ( f ˆ ( i − j ) ) 1 ≤ i , j ≤ n be the Toeplitz matrix generated by f. Then μ f ( C [ ε ] ) = det ( D n ( 2 ε − 1 ) T n ( f ) + D n ( 1 − ε ) ) = det A n ( C [ ε ] ) , where A n ( C [ ε ] ) is the n × n matrix A n ( C [ ε ] ) = ( 2 ε 1 − 1 ) f ˆ ( 0 ) + ( 1 − ε 1 ) ⋯ ( 2 ε 1 − 1 ) f ˆ ( 1 − n ) ( 2 ε 2 − 1 ) f ˆ ( 1 ) ⋯ ( 2 ε 2 − 1 ) f ˆ ( 2 − n ) ⋮ ⋱ ⋮ ( 2 ε n − 1 ) f ˆ ( n − 1 ) ⋯ ( 2 ε n − 1 ) f ˆ ( 0 ) + ( 1 − ε n ) .

Let ( ξ n ) n ∈ Z be the coordinate process on Ω = { 0 , 1 } Z , that is, ξ n ( x ) = x n . For integers n ≤ m, define F n m : = ∨ ( ξ n , ξ n + 1 , … , ξ m ) , the sigma-algebra generated by the coordinate maps ξ n , … , ξ m . Similarly, set F − ∞ n : = ∨ ( … , ξ n − 1 , ξ n ) , F n + ∞ : = ∨ ( ξ n , ξ n + 1 , … ) .

Then, for each ℓ ≥ 1, the ψ–mixing coefficient of μ f is defined by ψ μ f ( ℓ ) = sup A ∈ F − ∞ 0 , B ∈ F ℓ + ∞ μ f ( A ) μ f ( B ) > 0 μ f ( A ∩ B ) μ f ( A ) μ f ( B ) − 1 . We say that μ f is ψ–mixing if lim ℓ → ∞ ψ μ f ( ℓ ) = 0 . Theorem 2

Let f : S 1 → [ 0 , 1 ] be an integrable function which is not identically 0 or 1. Then for every integer ℓ ≥ 1, the ψ–mixing coefficient of the stationary determinantal process μ f satisfies (7) ψ μ f ( ℓ ) ≥ 1 − exp − ∑ n = ℓ + 1 ∞ | n | | f ˆ ( n ) | 2 . In particular, if μ f is ψ–mixing, then f ∈ H 1 / 2 ( T ).

Conversely, if there exists a constant τ > 0 such that (8) f ∈ H 1 / 2 ( T ) and τ ≤ f ( t ) ≤ 1 − τ for all t ∈ T , then μ f is ψ–mixing and its ψ–function satisfies the upper bound (9) ψ μ f ( ℓ ) ≤ 2 τ 2 ∑ n = ℓ + 1 ∞ | n | | f ˆ ( n ) | 2 exp 1 + 2 τ 2 ∑ n = ℓ + 1 ∞ | n | | f ˆ ( n ) | 2 .

We formulate the main results of our work. The first result gives a uniform exponential estimate for cylinder probabilities. It shows that, under the non-degeneracy condition τ ≤ f ≤ 1 − τ, the measure of every n-cylinder decays exponentially in n. This estimate will be used repeatedly to control the size of cylinder targets.

The next result is a conditional version of the previous cylinder estimate. It says that fixing additional coordinates decreases the measure at an exponential rate depending only on the number of newly fixed coordinates. This is the determinantal analogue of the finite-energy property for Gibbs measures.

We now state the main dynamical Borel–Cantelli result. The geometric assumption on the supports is the D-nested condition, while the probabilistic input is the quantitative mixing of the stationary determinantal measure. Together they imply the Sprindzuk condition and hence the strong Borel–Cantelli property. Theorem 6.

Let f : S 1 → [ 0 , 1 ] be a Borel function, and let ⟨ { 0 , 1 } Z , B , μ f , σ ⟩ be the dynamical system generated by the stationary determinantal point process associated with f, where μ f is the σ-invariant determinantal probability measure induced by the convolution kernel K f ( n , m ) = f ˆ ( n − m ) , n , m ∈ Z , and σ is the shift map.

Assume that there exist constants C , a > 0 and τ ∈ ( 0 , 1 / 2 ) such that | f ˆ ( n ) | ≤ C e − a | n | ( n ∈ Z ) , τ ≤ f ( t ) ≤ 1 − τ for a.e. t ∈ T . Suppose that there exists a fixed constant D ∈ N 0 such that the cylinders C L n ( ω ), n ≥ 1, are supported on finite intervals L n = [ m ( n ) , k ( n ) ] ⊂ Z and every pair L m , L n is D-nested in the sense of Definition 3. Then, 1.

the sequence of cylinders { C L n ( ω ) } n ≥ 1 satisfies (SP);

We first prove the uniform cylinder estimate. The key point is that the Shirai–Takahashi formula allows us to express each cylinder probability as a finite Toeplitz determinant. The non-degeneracy condition τ ≤ f ≤ 1 − τ then gives a uniform bound on the corresponding finite-dimensional operator. Proof.

Fix n ≥ 1 and a binary word ε = ( ε 1 , … , ε n ) ∈ { 0 , 1 } n . Consider the corresponding cylinder set C [ ε ] : = { x ∈ { 0 , 1 } Z : x 1 = ε 1 , … , x n = ε n } . By shift-invariance of the measure μ f , it suffices to consider cylinders supported on the index set { 1 , … , n }.

Let T n ( f ) : = ( f ˆ ( i − j ) ) 1 ≤ i , j ≤ n be the Toeplitz matrix generated by f. For a vector v ∈ C n , denote by D n ( v ) the diagonal matrix with diagonal entries v 1 , … , v n .

Using the determinantal representation of cylinder measures, we have (10) μ f ( C [ ε ] ) = det ( D n ( 2 ε − 1 ) T n ( f ) + D n ( 1 − ε ) ) , where 1 = ( 1 , … , 1 ) ∈ R n .

We first rewrite the matrix in (10) into a more convenient form. Set η : = 2 ε − 1 ∈ { − 1 , 1 } n , g : = 2 f − 1 . Then the following identity holds: (11) D n ( 2 ε − 1 ) T n ( f ) + D n ( 1 − ε ) = 1 2 ( I n + D n ( η ) T n ( g ) ) , where I n denotes the n × n identity matrix.

Indeed, since T n ( f ) = 1 2 ( T n ( g ) + I n ) , and 1 − ε i = 1 2 ( 1 − η i ) , i = 1 , … , n , we obtain D n ( 1 − ε ) = 1 2 ( I n − D n ( η ) ) , D n ( 2 ε − 1 ) = D n ( η ) . Substituting these expressions into the left-hand side of (11), we compute D n ( 2 ε − 1 ) T n ( f ) + D n ( 1 − ε ) = D n ( η ) · 1 2 ( T n ( g ) + I n ) + 1 2 ( I n − D n ( η ) ) = 1 2 ( D n ( η ) T n ( g ) + I n ) , which proves (11).

Combining (10) and (11), we obtain (12) μ f ( C [ ε ] ) = det ( 2 − 1 ( I n + D n ( η ) T n ( g ) ) ) = 2 − n det ( I n + A ) , where A : = D n ( η ) T n ( g ) .

We now estimate det ( I n + A ). Since D n ( η ) is diagonal with entries ± 1, we have ‖ D n ( η ) ‖ = 1 . Therefore, (13) ‖ A ‖ = ‖ D n ( η ) T n ( g ) ‖ = ‖ T n ( g ) ‖ . The matrix T n ( g ) is a finite compression of the bi-infinite Toeplitz operator T ( g ) acting on ℓ 2 ( Z ), hence ‖ T n ( g ) ‖ ≤ ‖ T ( g ) ‖ . Moreover, T ( g ) is unitarily equivalent to the multiplication operator by g on L 2 ( T ) (see, for example, [2], p. 179, Corollary 7.8), which yields ‖ T ( g ) ‖ = ‖ g ‖ ∞ . Using the assumption τ ≤ f ≤ 1 − τ almost everywhere, we obtain ‖ g ‖ ∞ = ‖ 2 f − 1 ‖ ∞ ≤ 1 − 2 τ < 1 . Consequently, (14) ‖ A ‖ ≤ 1 − 2 τ < 1 .

Let λ 1 ( A ) , … , λ n ( A ) denote the eigenvalues of A. Since the spectral radius satisfies ρ ( A ) ≤ ‖ A ‖, it follows from (14) that | λ i ( A ) | ≤ 1 − 2 τ , i = 1 , … , n . The eigenvalues of I n + A are 1 + λ i ( A ), and hence det ( I n + A ) = ∏ i = 1 n ( 1 + λ i ( A ) ) . By the triangle inequality, 1 − | λ i ( A ) | ≤ | 1 + λ i ( A ) | ≤ 1 + | λ i ( A ) | , i = 1 , … , n . Combining these inequalities, we obtain 2 n τ n ≤ | det ( I n + A ) | ≤ 2 n ( 1 − τ ) n . By (12), we have τ n ≤ μ f ( C [ ε ] ) ≤ ( 1 − τ ) n . This completes the proof of Theorem 4.  □

We next prove a conditional version of the previous estimate. This result shows that the measure of a cylinder decreases exponentially when additional coordinates are fixed. Such a bound is needed later when two shifted cylinders overlap, since their intersection is again a cylinder supported on the union of the two intervals. Proof.

Let Λ = ε 1 ⋯ ε m ∈ { 0 , 1 } m .

We present the proof in the two pure cases, namely when the defining word consists only of ones or only of zeros. In these cases the matrices M Λ and M Λ 1 are Hermitian positive definite, and the proof follows directly from the Cauchy interlacing theorem.

Define M Λ : = D m ( 2 ε − 1 ) T m ( f ) + D m ( 1 − ε ) , where T m ( f ) is the m × m Toeplitz matrix generated by f. By the Shirai–Takahashi formula for stationary determinantal processes, μ f ( [ Λ ] ) = det ( M Λ ) . Let Λ 1 be an extension of Λ: Λ 1 = ( ε 1 ( 1 ) , … , ε m + k ( 1 ) ) = ( ε 1 l , … , ε k 1 l ) ︸ Λ l ( ε 1 , … , ε m ) ︸ Λ ( ε 1 r , … , ε k 2 r ) ︸ Λ r , where k 1 + k 2 = k. Set A : = D k 1 ( 2 ε l − 1 ) , B : = D m ( 2 ε − 1 ) , C : = D k 2 ( 2 ε r − 1 ) , and E l : = D k 1 ( 1 − ε l ) , E : = D m ( 1 − ε ) , E r : = D k 2 ( 1 − ε r ) . Then the matrix M Λ 1 has the block form M Λ 1 = A T l l + E l A T l A T l r B T m l B T + E B T m r C T r l C T r C T r r + E r , with B T + E = M Λ . Consequently, μ f ( [ Λ 1 ] ) = det ( M Λ 1 ) , and therefore μ f ( C 1 ) μ f ( C ) = det ( M Λ 1 ) det ( M Λ ) . By Theorem 4, τ m + k ≤ det ( M Λ 1 ) ≤ ( 1 − τ ) m + k . Moreover, (15) τ ≤ λ 1 ( Λ 1 ) ≤ ⋯ ≤ λ m + k ( Λ 1 ) ≤ 1 − τ . Since M Λ is a principal submatrix of M Λ 1 and both matrices are Hermitian, the Cauchy interlacing theorem (Lemma 4) yields λ i ( Λ 1 ) ≤ λ i ( Λ ) ≤ λ i + k ( Λ 1 ) , i = 1 , … , m . Using the product representation of determinants, μ f ( C 1 ) μ f ( C ) = ∏ i = 1 m + k λ i ( Λ 1 ) ∏ i = 1 m λ i ( Λ ) ≥ ∏ i = 1 k λ i ( Λ 1 ) ≥ τ k . Similarly, μ f ( C 1 ) μ f ( C ) ≤ ∏ i = m + 1 m + k λ i ( Λ 1 ) ≤ ( 1 − τ ) k . Since k = | Λ 1 | − | Λ |, we obtain τ | Λ 1 | − | Λ | ≤ μ f ( C 1 ) μ f ( C ) ≤ ( 1 − τ ) | Λ 1 | − | Λ | .

It remains to consider the mixed case, when both symbols 0 and 1 occur in the word ε. In this case the matrix M Λ = D Λ ( 2 ε − 1 ) T Λ ( f ) + D Λ ( 1 − ε ) is not Hermitian in general. Put S Λ : = D Λ ( 2 ε − 1 ) . Then H Λ : = S Λ M Λ = T Λ ( f ) − D Λ ( 1 − ε ) is Hermitian. Similarly, H Λ 1 : = S Λ 1 M Λ 1 = T Λ 1 ( f ) − D Λ 1 ( 1 − ε 1 ) is Hermitian. Since C Λ 1 ( ε 1 ) ⊂ C Λ ( ε ), the matrix H Λ is a principal submatrix of H Λ 1 . Hence the Cauchy interlacing theorem applies. Moreover, det M Λ = det S Λ det H Λ , det M Λ 1 = det S Λ 1 det H Λ 1 . Thus the signs coming from the zero symbols are completely accounted for by det S Λ and det S Λ 1 . Therefore, applying the same eigenvalue-ratio argument as above together with the Cauchy interlacing theorem, we obtain τ | Λ 1 | − | Λ | ≤ μ f ( C Λ 1 ( ε 1 ) ) μ f ( C Λ ( ε ) ) ≤ ( 1 − τ ) | Λ 1 | − | Λ | .

This completes the proof of Theorem 5.  □