The purpose of this paper is to investigate efficient representations of the residue classes modu... more The purpose of this paper is to investigate efficient representations of the residue classes moduloq, by performing sum and product set operations starting from a given subset A of Zq . We consider the case of very small sets A and compositeq for which not much seemed known (non- trivial results were recently obtained when q is prime or when log|A| logq). Roughly speaking we show that all residue classes are obtained from ak-fold sum of anr-fold product set ofA, where r logq and logk logq, provided the residue sets q0(A) are large for all large divisors q 0 of q. Even in the special case of prime modulus q, some results are new, when considering large but bounded sets A. It follows for instance from our estimates that one can obtain r as small as r logq/ log|A| with similar restriction on k, something not covered by earlier work of Konya- gin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking...
We obtain various bounds on orbit length of modular reductions of algebraic dynamical systems gen... more We obtain various bounds on orbit length of modular reductions of algebraic dynamical systems generated by polynomials with integer coefficients. In particular we extend a recent result of Chang (2015) in two different directions.
In this talk I will present estimates on incomplete character sums in finite fields, with special... more In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particular, we improve on earlier work of Davenport-Lewis and Karacuba.
We show that several statistics of the number of intersections between random eigenfunctions of g... more We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve in flat tori are universal under various families of randomness.
The purpose of this paper is to investigate efficient representations of the residue classes modu... more The purpose of this paper is to investigate efficient representations of the residue classes moduloq, by performing sum and product set operations starting from a given subset A of Zq . We consider the case of very small sets A and compositeq for which not much seemed known (non- trivial results were recently obtained when q is prime or when log|A| logq). Roughly speaking we show that all residue classes are obtained from ak-fold sum of anr-fold product set ofA, where r logq and logk logq, provided the residue sets q0(A) are large for all large divisors q 0 of q. Even in the special case of prime modulus q, some results are new, when considering large but bounded sets A. It follows for instance from our estimates that one can obtain r as small as r logq/ log|A| with similar restriction on k, something not covered by earlier work of Konya- gin and Shparlinski. On the technical side, essential use is made of Freiman's structural theorem on sets with small doubling constant. Taking...
We obtain various bounds on orbit length of modular reductions of algebraic dynamical systems gen... more We obtain various bounds on orbit length of modular reductions of algebraic dynamical systems generated by polynomials with integer coefficients. In particular we extend a recent result of Chang (2015) in two different directions.
In this talk I will present estimates on incomplete character sums in finite fields, with special... more In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particular, we improve on earlier work of Davenport-Lewis and Karacuba.
We show that several statistics of the number of intersections between random eigenfunctions of g... more We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve in flat tori are universal under various families of randomness.
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Papers by Mei-chu Chang