Caps and progression-free sets in $${{\mathbb {Z}}}_m^n$$
Abstract: We study progression-free sets in the abelian groups $$G=({{\mathbb {Z}}}_m^n,+)$$ G = ( Z m n , + ) . Let $$r_k({{\mathbb {Z}}}_m^n)$$ r k ( Z m n ) denote the maximal size of a set $$S \subset {{\mathbb {Z}}}_m^n$$ S ⊂ Z m n that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that $$r_3({{\mathbb {Z}}}_m^n) \ge C_m \frac{((m+2)/2)^n}{\sqrt{n}}$$ r 3 ( Z m n ) ≥ C m ( ( m + 2 ) / 2 ) n n , when m is even… Show more
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Cited by 7 publications
References 53 publications
For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in $$(\mathbb {Z}_{m}^{n},+)$$ ( Z m n , + ) . Let $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) denote the maximal size of a subset of $$\mathbb {Z}_{m}^{n}$$ Z m n without arithmetic progressions of length k and let $$P^{-}(m)$$ P - ( m ) denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) : If $$k\ge 5$$ k ≥ 5 is odd and $$P^{-}(m)\ge (k+2)/2$$ P - ( m ) ≥ ( k + 2 ) / 2 , then $$\begin{aligned} r_k(\mathbb {Z}_m^n) \gg _{m,k} \frac{\bigl \lfloor \frac{k-1}{k+1}m +1\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 1 k + 1 m + 1 ⌋ n n ⌊ k - 1 k + 1 m ⌋ / 2 . If $$k\ge 4$$ k ≥ 4 is even, $$P^{-}(m) \ge k$$ P - ( m ) ≥ k and $$m \equiv -1 \bmod k$$ m ≡ - 1 mod k , then $$\begin{aligned} r_{k}(\mathbb {Z}_{m}^{n}) \gg _{m,k} \frac{\bigl \lfloor \frac{k-2}{k}m + 2\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 2 k m + 2 ⌋ n n ⌊ k - 2 k m + 1 ⌋ / 2 . Moreover, we give some further improved lower bounds on $$r_k(\mathbb {Z}_p^n)$$ r k ( Z p n ) for primes $$p \le 31$$ p ≤ 31 and progression lengths $$4 \le k \le 8$$ 4 ≤ k ≤ 8 .
For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in $$(\mathbb {Z}_{m}^{n},+)$$ ( Z m n , + ) . Let $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) denote the maximal size of a subset of $$\mathbb {Z}_{m}^{n}$$ Z m n without arithmetic progressions of length k and let $$P^{-}(m)$$ P - ( m ) denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for $$r_{k}(\mathbb {Z}_{m}^{n})$$ r k ( Z m n ) : If $$k\ge 5$$ k ≥ 5 is odd and $$P^{-}(m)\ge (k+2)/2$$ P - ( m ) ≥ ( k + 2 ) / 2 , then $$\begin{aligned} r_k(\mathbb {Z}_m^n) \gg _{m,k} \frac{\bigl \lfloor \frac{k-1}{k+1}m +1\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-1}{k+1}m \rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 1 k + 1 m + 1 ⌋ n n ⌊ k - 1 k + 1 m ⌋ / 2 . If $$k\ge 4$$ k ≥ 4 is even, $$P^{-}(m) \ge k$$ P - ( m ) ≥ k and $$m \equiv -1 \bmod k$$ m ≡ - 1 mod k , then $$\begin{aligned} r_{k}(\mathbb {Z}_{m}^{n}) \gg _{m,k} \frac{\bigl \lfloor \frac{k-2}{k}m + 2\bigr \rfloor ^{n}}{n^{\lfloor \frac{k-2}{k}m + 1\rfloor /2}}. \end{aligned}$$ r k ( Z m n ) ≫ m , k ⌊ k - 2 k m + 2 ⌋ n n ⌊ k - 2 k m + 1 ⌋ / 2 . Moreover, we give some further improved lower bounds on $$r_k(\mathbb {Z}_p^n)$$ r k ( Z p n ) for primes $$p \le 31$$ p ≤ 31 and progression lengths $$4 \le k \le 8$$ 4 ≤ k ≤ 8 .
Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$
Let us fix a prime p and a homogeneous system of m linear equations aj,1x1 + • • • + a j,k x k = 0 for j = 1, . . . , m with coefficients aj,i ∈ Fp. Suppose that k ≥ 3m, that aj,1 + • • • + a j,k = 0 for j = 1, . . . , m and that every m × m minor of the m × k matrix (aj,i)j,i is non-singular. Then we prove that for any (large) n, any subset A ⊆ F n p of size |A| > C • Γ n contains a solution (x1, . . . , x k ) ∈ A k to the given system of equations such that the vectors x1, . . . , x k ∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ < p.The crucial point here is the condition for the vectors x1, . . . , x k in the solution (x1, . . . , x k ) ∈ A k to be distinct. If we relax this condition and only demand that x1, . . . , x k are not all equal, then the statement would follow easily from Tao's slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.
Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$
Let us fix a prime p and a homogeneous system of m linear equations $$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ a j , 1 x 1 + ⋯ + a j , k x k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m with coefficients $$a_{j,i}\in \mathbb {F}_p$$ a j , i ∈ F p . Suppose that $$k\ge 3m$$ k ≥ 3 m , that $$a_{j,1}+\dots +a_{j,k}=0$$ a j , 1 + ⋯ + a j , k = 0 for $$j=1,\dots ,m$$ j = 1 , ⋯ , m and that every $$m\times m$$ m × m minor of the $$m\times k$$ m × k matrix $$(a_{j,i})_{j,i}$$ ( a j , i ) j , i is non-singular. Then we prove that for any (large) n, any subset $$A\subseteq \mathbb {F}_p^n$$ A ⊆ F p n of size $$|A|> C\cdot \Gamma ^n$$ | A | > C · Γ n contains a solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to the given system of equations such that the vectors $$x_1,\dots ,x_k\in A$$ x 1 , ⋯ , x k ∈ A are all distinct. Here, C and $$\Gamma $$ Γ are constants only depending on p, m and k such that $$\Gamma <p$$ Γ < p . The crucial point here is the condition for the vectors $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k in the solution $$(x_1,\dots ,x_k)\in A^k$$ ( x 1 , ⋯ , x k ) ∈ A k to be distinct. If we relax this condition and only demand that $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.
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