On a Lemma Arising in the Solution of Waring’s Problem

Siberian Mathematical Journal - We give a simple proof of one assertion used in solving Waring’s problem.     

Convergence of subsequences of partial cubic sums of Fourier series in mean and almost everywhere

Under certain assumptions on the regularity of a function necessary and sufficient conditions are found for under which the integrability of (¦f¦) implies, for every function f measurable onT ^d, the existence of a subsequence of cubic sums of the Fourier series of f that converges to f in mean or almost everywhere.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 63–77, September, 1992.     

Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ₁,θ₂,…,θ_N−1} is an N-spectrum for A(x) if the θ_j are all distinct and

     

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x ₁ + s)... (x _ν + s) ≡ (y ₁ + s)... (x _ν + s) ? 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.     

Multiplicative congruences with variables from short intervals

Recently, several bounds have been obtained on the number of solutions of congruences of the type $$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$ where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums     

On the number of popular differences

We prove that there exists an absolute constant c > 0 such that for any finite set A ⊆ ℤ with |A| ≥ 2 and any positive integer m < c|A|/ ln |A| there are at most m integers b > 0 satisfying |(A + b) \ A| ≤ m; equivalently, there are at most m positive integers possessing |A| −m (or more) representations as a difference of two elements of A.