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Papers by hoi nguyen

Research paper thumbnail of On the Number of Real Roots of Random Polynomials

Communications in Contemporary Mathematics, 2015

Roots of random polynomials have been studied intensively in both analysis and probability for a ... more Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is [Formula: see text]. In this paper, we determine the true nature of the error term by showing that the expectation equals [Formula: see text]. Prior to this paper, the error term [Formula: see text] has been known only for polynomials with Gaussian coefficients.

Research paper thumbnail of A new approach to an old problem of Erdős and Moser

Journal of Combinatorial Theory, Series A, 2012

Let ηi, i = 1,. .. , n be iid Bernoulli random variables, taking values ±1 with probability 1 2. ... more Let ηi, i = 1,. .. , n be iid Bernoulli random variables, taking values ±1 with probability 1 2. Given a multiset V of n elements v1,. .. , vn of an additive group G, we define the concentration probability of V as ρ(V) := sup v∈G P(η1v1 +. .. ηnvn = v). An old result of Erdős and Moser asserts that if vi are distinct real numbers then ρ(V) is O(n − 3 2 log n). This bound was then refined by Sárközy and Szemerédi to O(n − 3 2), which is sharp up to a constant factor. The ultimate result dues to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal concentration probability; the result now becomes classic in algebraic combinatorics. In this paper, we will prove that the optimal sets from Stanley's work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.

Research paper thumbnail of Eigenvectors of random matrices of symmetric entry distributions

Proceedings of the American Mathematical Society

In this short note we study a non-degeneration property of eigenvectors of symmetric random matri... more In this short note we study a non-degeneration property of eigenvectors of symmetric random matrices with entries of symmetric sub-gaussian distributions. Our result is asymptotically optimal under the sub-exponential regime.

Research paper thumbnail of Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

Let MnM_nMn denote a random symmetric nnn by nnn matrix, whose upper diagonal entries are iid Berno... more Let MnM_nMn denote a random symmetric nnn by nnn matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that MnM_nMn is non-singular with probability 1−O(n−C)1-O(n^{-C})1−O(n−C) for any positive constant CCC. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.

Research paper thumbnail of Classification theorems for sumsets modulo a prime

Journal of Combinatorial Theory, Series A, 2009

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classi... more Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of A ? (2) When can one represent every element of Zp as a sum of some elements of A ? (3) When can one represent every element of Zp as a sum of l elements of A ?

Research paper thumbnail of On the Number of Real Roots of Random Polynomials

Communications in Contemporary Mathematics, 2015

Roots of random polynomials have been studied intensively in both analysis and probability for a ... more Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is [Formula: see text]. In this paper, we determine the true nature of the error term by showing that the expectation equals [Formula: see text]. Prior to this paper, the error term [Formula: see text] has been known only for polynomials with Gaussian coefficients.

Research paper thumbnail of A new approach to an old problem of Erdős and Moser

Journal of Combinatorial Theory, Series A, 2012

Let ηi, i = 1,. .. , n be iid Bernoulli random variables, taking values ±1 with probability 1 2. ... more Let ηi, i = 1,. .. , n be iid Bernoulli random variables, taking values ±1 with probability 1 2. Given a multiset V of n elements v1,. .. , vn of an additive group G, we define the concentration probability of V as ρ(V) := sup v∈G P(η1v1 +. .. ηnvn = v). An old result of Erdős and Moser asserts that if vi are distinct real numbers then ρ(V) is O(n − 3 2 log n). This bound was then refined by Sárközy and Szemerédi to O(n − 3 2), which is sharp up to a constant factor. The ultimate result dues to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal concentration probability; the result now becomes classic in algebraic combinatorics. In this paper, we will prove that the optimal sets from Stanley's work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.

Research paper thumbnail of Eigenvectors of random matrices of symmetric entry distributions

Proceedings of the American Mathematical Society

In this short note we study a non-degeneration property of eigenvectors of symmetric random matri... more In this short note we study a non-degeneration property of eigenvectors of symmetric random matrices with entries of symmetric sub-gaussian distributions. Our result is asymptotically optimal under the sub-exponential regime.

Research paper thumbnail of Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

Let MnM_nMn denote a random symmetric nnn by nnn matrix, whose upper diagonal entries are iid Berno... more Let MnM_nMn denote a random symmetric nnn by nnn matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that MnM_nMn is non-singular with probability 1−O(n−C)1-O(n^{-C})1−O(n−C) for any positive constant CCC. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.

Research paper thumbnail of Classification theorems for sumsets modulo a prime

Journal of Combinatorial Theory, Series A, 2009

Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classi... more Let Zp be the finite field of prime order p and A be a subsequence of Zp. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of A ? (2) When can one represent every element of Zp as a sum of some elements of A ? (3) When can one represent every element of Zp as a sum of l elements of A ?